Surface acoustic wave resonator, surface acoustic wave oscillator, and electronic apparatus

ABSTRACT

A surface acoustic wave resonator includes an IDT that is disposed on a quartz crystal substrate of Euler angles (−1.5°≦φ≦1.5°, 117°≦θ≦142°, ψ) and excites a surface acoustic wave resonant in an upper part of a stop-band of the IDT, and inter-electrode finger grooves that are acquired by depressing the substrate located between electrode fingers configuring the IDT. The wavelength of the surface acoustic wave, the depth of the inter-electrode finger grooves, the line occupancy ratio of the IDT, and the film thickness of the electrode fingers of the IDT are set in correspondence with one another.

CROSS-REFERENCE TO RELATED APPLICATIONS

This is a continuation patent application of U.S. application Ser. No.13/162,162 filed Jun. 16, 2011, now U.S. Pat. No. 9,088,263 issued Jul.21, 2015, which claims priority to Japanese Patent Application No.2010-138495, filed Jun. 17, 2010, all of which are expresslyincorporated by reference herein in their entireties.

BACKGROUND

1. Technical Field

The present invention relates to a surface acoustic wave resonator and asurface acoustic wave oscillator in which the surface acoustic waveresonator is mounted, and more particularly, to a type of surfaceacoustic wave resonator in which grooves are arranged in a substratesurface and a surface acoustic wave oscillator and an electronicapparatus in which such a type of surface acoustic wave resonator ismounted.

2. Related Art

In a surface acoustic wave (SAW) device (for example, a SAW resonator),the influence of the stop band of a SAW, a cut angle of a piezoelectricsubstrate (for example, a quartz crystal substrate), the formed shape ofan IDT (inter-digital transducer), and the like on the change in thefrequency-temperature characteristics is great.

For example, in Japanese Patent No. 3266846 (JP-A-11-214958), aconfiguration used for exciting an upper end mode and a lower end modeof a stop band of the SAW, distributions of standing waves in the upperend mode and the lower end mode of the stop band, and the like aredisclosed.

In addition, in JP-A-2002-100959, JP-A-2006-148622, JP-A-2007-208871,JP-A-2007-267033, and JP-A-2007-300287, the frequency-temperaturecharacteristics of the stop band upper end mode of the SAW that arebetter than those of the stop band lower end mode are described.

Among them, in JP-A-2002-100959, the frequency-temperaturecharacteristics are described to be improved by using the resonance ofthe upper end of the stop band using a rotated Y-cut X-propagationcrystal quartz substrate more than a case where the resonance of thelower end of the stop band is used.

In addition, in JP-A-2006-148622 and JP-A-2007-208871, in order toacquire good frequency-temperature characteristics from a SAW deviceusing a Rayleigh wave, adjusting the cut angle of the quartz crystalsubstrate and thickening the normalized electrode film thickness (H/λ)up to 0.1 are described to be performed. Here, λ is the wavelength ofthe SAW.

Furthermore, in JP-A-2007-267033, adjusting the cut angle of the quartzcrystal substrate and thickening the normalized electrode film thickness(H/λ) so as to be equal to or greater than 0.045 are described to beperformed for a SAW device using a Rayleigh wave.

In JP-B-2-7207 (JP-A-57-5418) and “Manufacturing Conditions andCharacteristics of Groove-Type SAW Resonator” (The Institute ofElectronics, Information and Communication Engineers Technical ResearchReport MW82-59 (1982)), in a SAW device using an ST-cut quartz crystalsubstrate, grooves are described to be arranged between electrodefingers configuring the IDT and between conductive strips configuring areflector. In addition, in “Manufacturing Conditions and Characteristicsof Groove-Type SAW Resonator” described above, the frequency-temperaturecharacteristics are described to change in accordance with the depth ofthe grooves.

In addition, also in JP-A-2-189011, JP-A-5-90865, JP-A-1-231412, andJP-A-61-92011, the frequency characteristics are described to beadjusted by forming grooves in a piezoelectric substrate formed fromquartz crystal or the like.

Furthermore, it is disclosed in JP-A-10-270974 that, in atransversal-type SAW filer, grooves are formed by etching and processingthe surface of a piezoelectric substrate between electrodes of an IDT,and by forming electrode fingers of pure metal or an alloy that hasspecific gravity higher than that of aluminum, the appearing propagationspeed is reduced so as to decrease the pitch of the electrode fingers,whereby realizing the miniaturization of a corresponding chip.

In JP-B-1-34411, implementation of third-order frequency-temperaturecharacteristics by exciting a SSBW (Surface Skimming Bulk Wave) in a SAWresonator that is formed by forming an IDT electrode, of which anormalized electrode film thickness (H/λ) is in the range of2.0≦H/λ≦4.0%, from aluminum in a quartz crystal substrate having arotated Y-cut, a cut angle of −43° or −52°, and the slip wavepropagation direction set in the Z′-axis direction (Euler angles (φ, θ,ψ)=(0°, 38°≦θ≦47°, 90°)) is disclosed. However, this SAW resonator hasfeatures in that an SH wave propagating right below the surface of apiezoelectric substrate is excited by an IDT, and the vibrational energyis confined right below the electrode, and accordingly, the SH wave isbasically a wave that progresses inside the substrate. Therefore, thereflection efficiency of the SAW using a grating reflector is lower thanthat of an ST-cut quartz crystal SAW propagating along the surface ofthe piezoelectric substrate, and there is a problem in that aminiaturized SAW device having a high Q value cannot be easilyimplemented.

In addition, in PCT Republication No. WO2005/099089 A1, in order tosolve the above-described problems, a SAW device acquired by forming anIDT electrode and a grating reflector on the surface of a quartz crystalsubstrate having Euler angles (φ, θ, ψ)=(0°, −64°<θ<−49.3°, 85°≦ψ≦95°)is proposed.

Furthermore, in JP-A-2006-203408, in consideration of a problem in thata Q value or the frequency stability deteriorates due to stressmigration that occurs due to a large electrode film thickness, it isdisclosed that grooves are formed on a crystal quartz substrate locatedin an area corresponding to a space between electrode fingers through anetching process, and when the depth of the grooves is H_(p), and thefilm thickness of the metal film is H_(m), a normalized electrode filmthickness (H/λ) is set in the range of 0.04<H/λ<0.12 (here,H=H_(p)+H_(m)). Accordingly, a SAW device, in which the variation in thefrequency is suppressed, having a high Q value can be realized.

In JP-A-2009-225420, in a SAW device using a so-called in-plane rotationST quartz crystal substrate, disclosed in JP-A-2006-148622,JP-A-2007-208871, JP-A-2007-267033, and JP-A-2007-300287, consideringthat side etching progresses in a process of forming electrode fingersthrough etching due to a large electrode film thickness, an individualline occupancy ratio varies, and the amount of change in the frequencyis large when the temperature changes, so as to cause serious problemsin the reliability and the quality of the product, it is proposed to usean in-plane rotation ST-cut quartz crystal having Euler angles (φ, θ,ψ)=(0°, 95°≦θ≦155°, 33°≦|ψ|≦46°). By using this quartz crystalsubstrate, by exciting the upper limit mode of the stop band of thesurface acoustic wave, a SAW device suppressing the unbalance in thefrequency variation can be implemented.

However, although the unbalance of the variation in the frequency can besuppressed while securing an effective film thickness by forming thegrooves by etching the surface of the quartz crystal substrate betweenthe electrode fingers as above, the frequency-temperaturecharacteristics in the operating temperature range of the SAW devicestill have second-order characteristics, and the width of the variationin the frequency is not decreased much.

In addition, in Japanese Patent No. 3851336, while a configuration forforming a curve representing the frequency-temperature characteristicsas a third-order curve in a SAW device using an LST-cut quartz crystalsubstrate is described, a substrate having a cut angle representing thetemperature characteristics as represented by a third-order curve isdescribed to have not been found in a SAW device using a Rayleigh wave.

As described above, factors for improving the frequency-temperaturecharacteristics include many things, and, particularly in a SAW deviceusing a Rayleigh wave, forming the film thickness of the electrodeconfiguring the IDT to be large is considered as one factor contributingto the frequency-temperature characteristics. However, the applicants ofthis application has found through experiments that, when the filmthickness of the electrode is formed to be large, environment-resistantcharacteristics such as characteristics that change by time ortemperature-resistant shock characteristics deteriorate. In addition, ina case where the main purpose is to improve the frequency-temperaturecharacteristics, the film thickness of the electrode needs to be largeas described above, and it is difficult to avoid the deterioration ofthe characteristics changing by time and the temperature-resistant shockcharacteristics that is accompanied with the large film thickness. Thiscan be also applied to the Q value, and it is difficult to realize ahigh Q value without forming the film thickness of the electrode to belarge. In addition, by forming the film thickness of the electrode to belarge, a CI (crystal impedance) value also increases, whereby thestability of oscillation is degraded.

Therefore, aspects of the invention for providing a surface acousticwave resonator, a surface acoustic wave oscillator, and an electronicapparatus are, first, to realize good frequency-temperaturecharacteristics, second, to improve the environment-resistantcharacteristics, third, to acquire a high Q value, and, fourth, toacquire a low CI value.

SUMMARY

An advantage of some aspects of the invention is that it provides asurface acoustic wave resonator having superior stability of oscillationregardless of use environments, and a surface acoustic wave oscillatorand an electronic apparatus including the resonator.

Application Example 1

According to this application example of the invention, there isprovided a surface acoustic wave resonator including: an IDT that isdisposed on a quartz crystal substrate of Euler angles (−1.5°≦φ≦1.5°,117°≦θ≦142°, and 42.79°≦|ψ|≦49.57° and excites a surface acoustic waveof a stop-band upper end mode; inter-electrode finger grooves that areacquired by depressing the substrate located between electrode fingersconfiguring the IDT; and one pair of reflection units that are arrangedso as allow the IDT to be disposed therebetween in a propagationdirection of the surface acoustic wave and reflect the surface acousticwave. In a case where a wavelength of the surface acoustic wave is λ,and a depth of the inter-electrode finger grooves is G, “0.01λ≦G” issatisfied. In addition, in a case where a line occupancy ratio of theIDT is η, the depth G of the inter-electrode finger grooves and the lineoccupancy ratio η satisfy following relationships.−2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775, wherein 0.0100λ≦G≦0.0500λ−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775, wherein 0.0500λ<G≦0.0695λ

Accordingly, a surface acoustic wave resonator can be acquired in which,first, good frequency-temperature characteristics are realized, second,the environment-resistant characteristics are improved, third, a high Qvalue is acquired, and, fourth, a low CI value is acquired. In otherwords, a surface acoustic wave resonator having superior oscillationstability regardless of use environments can be acquired.

Application Example 2

In the above-described surface acoustic wave resonator, it is preferablethat the depth G of the inter-electrode finger grooves satisfies arelationship of “0.01λ≦G≦0.0695λ”.

In such a case, even in a case where the depth G of the inter-electrodefinger groove is mismatched due to an error during a manufacturingprocess, a surface acoustic wave resonator that suppresses theindividual shift of the resonance frequency to be within a correctablerange can be acquired.

Application Example 3

In the above-described surface acoustic wave resonator, it is preferablethat, when a film thickness of electrode fingers of the IDT is H, arelationship of “0<H≦0.035λ” is satisfied.

In such a case, a surface acoustic wave resonator having goodfrequency-temperature characteristics within the operating temperaturerange can be acquired. In addition, according to such a feature, thedeterioration of the environment-resistant characteristics accompanyingthe increase in the film thickness of the electrode can be suppressed.

Application Example 4

In the above-described surface acoustic wave resonator, it is preferablethat the line occupancy ratio η satisfies a relationship of“η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)”.

In such a case, the second-order temperature coefficient can becontrolled to be approximately within ±0.01 ppm/° C.².

Application Example 5

In the above-described surface acoustic wave resonator, it is preferablethat a sum of the depth G of the inter-electrode finger grooves and thefilm thickness H of the electrode fingers satisfies a relationship of“0.0407λ≦G+H”.

In such a case, a Q value that is higher than that of a general surfaceacoustic wave resonator can be acquired.

Application Example 6

In the above-described surface acoustic wave resonator, it is preferablethat the Euler angles ψ and θ satisfy a relationship of“ψ=1.191×10⁻³×θ³−4.490×10⁻¹×θ²+5.646×10¹×θ−2.324×10³±1.0”.

In such a case, a surface acoustic wave resonator having excellentfrequency-temperature characteristics in a broad range can be acquired.

Application Example 7

In the above-described surface acoustic wave resonator, it is preferablethat, when a frequency of the stop-band upper end mode in the IDT isft2, a frequency of a stop-band lower end mode in the reflection unitsis fr1, and a frequency of the stop-band upper end mode of the reflectoris fr2, a relationship of “fr1<ft2<fr2” is satisfied.

In such a case, the reflection coefficient |Γ| of the reflection unitincreases for the frequency ft2 of the stop band upper end mode of theIDT, and an excited surface acoustic wave of the stop band upper endmode from the IDT is reflected to the IDT side by the reflection unitwith a high reflection coefficient. Accordingly, energy confinement ofthe surface acoustic wave of the stop band upper end mode becomesstrong, and a surface acoustic wave resonator with low loss can berealized.

Application Example 8

In the above-described surface acoustic wave resonator, it is preferablethat the reflection units are arranged so as to be parallel to theelectrode fingers configuring the IDT and are configured in groovesacquired by depressing the quartz crystal substrate.

In such a case, the degree of easiness in manufacturing of thereflection unit can be raised. In addition, since the forming of theconductive strips is unnecessary, the characteristic variation of thereflection unit can be suppressed.

Application Example 9

In the above-described surface acoustic wave resonator, it is preferablethat the grooves included in the reflection units are grooves of aplurality of lines parallel to one another.

In such a case, the degree of easiness in manufacturing of thereflection unit can be raised. In addition, since the forming of theconductive strips is unnecessary, the characteristic variation of thereflection unit can be suppressed.

Application Example 10

In the above-described surface acoustic wave resonator, it is preferablethat, when a film thickness of the electrode fingers configuring the IDTis H_(mT), a depth of the inter-electrode finger grooves is H_(gT), aneffective film thickness of the electrode fingers is H_(T)/λ (here,H_(T)=H_(mT)+H_(gT)), and, when the depth of the grooves included in thereflection units is H_(gR), the IDT and the reflection units satisfy arelationship of “H_(T)/λ<H_(gR)/λ”.

In such a case, even in a case where the conductive strip is omitted,the reflection characteristics of the reflection unit are improved, andthe energy confinement effect of the SAW of the stop band upper end modebecomes more remarkable, whereby the Q value is further improved. Inaddition, since the effective film thickness of the electrode finger ofthe IDT relatively decreases, the electromechanical coupling coefficientof the IDT can be increased, whereby the CI value can be furtherdecreased.

Application Example 11

In the above-described surface acoustic wave resonator, it is preferablethat the depth H_(gR) of the grooves is equal to or more than 3λ.

In such a case, by only forming a groove having a large depth in themanufacturing of the reflection unit, the degree of easiness inmanufacturing thereof can be remarkably raised.

Application Example 12

In the above-described surface acoustic wave resonator, it is preferablethat the reflection unit is configured by conductive strips of aplurality of lines that are disposed on the quartz crystal substrate andparallel to each other.

In such a case, the degree of easiness in manufacturing of thereflection unit can be raised. In addition, since the forming of thegroove is unnecessary, the characteristic variation of the reflectionunit can be suppressed.

Application Example 13

In the above-described surface acoustic wave resonator, it is preferablethat, when a film thickness of the electrode fingers configuring the IDTis H_(mT), a depth of the inter-electrode finger grooves is H_(gT), aneffective film thickness of the electrode fingers is H_(T)/λ (here,H_(T)=H_(mT)+H_(gT)), and a film thickness of the conductive strips isH_(mR), the IDT and the reflection units satisfy a relationship of“H_(T)/λ<H_(mR)/λ”.

In such a case, even in a case where the groove is omitted, thereflection characteristics of the reflection unit are improved, and theenergy confinement effect of the SAW of the stop band upper end modebecomes more remarkable, whereby the Q value is further improved. Inaddition, since the effective film thickness of the electrode finger ofthe IDT relatively decreases, the electromechanical coupling coefficientof the IDT can be increased, whereby the CI value can be furtherdecreased.

Application Example 14

In the above-described surface acoustic wave resonator, it is preferablethat the reflection unit is configured by a plurality of lines of thegrooves that are parallel to each other and a plurality of lines ofconductive strips that are disposed on the quartz crystal substrate soas to be adjacent to the grooves and are parallel to each other.

In such a case, a surface acoustic wave resonator having superiorfrequency-temperature characteristics can be acquired.

Application Example 15

In the above-described surface acoustic wave resonator, it is preferablethat, when a film thickness of the electrode fingers configuring the IDTis H_(mT), a depth of the inter-electrode finger grooves is H_(gT), aneffective film thickness of the electrode fingers is H_(T)/λ (here,H_(T)=H_(mT)+H_(gT)), a film thickness of the conductive strips isH_(mR), a depth of the grooves included in the reflection unit isH_(gR), and an effective film thickness of the conductive strips isH_(R)/λ (here, H_(R)=H_(mR)+H_(gR)), the IDT and the reflection unitssatisfy a relationship of “H_(T)/λ<H_(R)/λ”.

In such a case, the reflection characteristics of the reflection unitare improved together with the increase in the effective film thicknessof the conductive strip, and the energy confinement effect of the SAW ofthe stop band upper end mode becomes more remarkable, whereby the Qvalue is further improved. In addition, since the effective filmthickness of the electrode finger of the IDT relatively decreases, theelectromechanical coupling coefficient of the IDT can be increased,whereby the CI value can be further decreased.

Application Example 16

In the above-described surface acoustic wave resonator, it is preferablethat the film thickness H_(mT) of the electrode fingers configuring theIDT and the film thickness H_(mR) of the conductive strips satisfy arelationship of “H_(mT)/λ=H_(mR)/λ”, and the depth H_(gT) of theinter-electrode finger grooves and the depth H_(gR) of the groovesincluded in the reflection unit satisfy a relationship of“H_(gT)/λ<H_(gR)/λ”.

In such a case, the implementation of a high Q value and theimplementation of low CI can be achieved together. In addition, aconductive film having a single film thickness may be formed, andaccordingly, the degree of easiness in the manufacturing is raised.

Application Example 17

In the above-described surface acoustic wave resonator, it is preferablethat the depth H_(gT) of the inter-electrode finger grooves and thedepth H_(gR) of the grooves included in the reflection unit satisfy arelationship of “H_(gT)/λ=H_(gR)/λ”, and the film thickness H_(mT) ofthe electrode fingers configuring the IDT and the film thickness H_(mR)of the conductive strips satisfy a relationship of “H_(mT)/λ<H_(mR)/λ”.

In such a case, the implementation of a high Q value and theimplementation of low CI can be achieved together. In addition, thegroove may be processed under one type of condition, and accordingly,the degree of easiness in the manufacturing is raised.

Application Example 18

In the above-described surface acoustic wave resonator, it is preferablethat the reflection unit is configured by an end face of the quartzcrystal substrate.

In such a case, the degree of easiness in the manufacturing of thereflection unit can be raised. In addition, since formation of a grooveor a conductive strip is unnecessary, the characteristic variation ofthe reflection unit can be suppressed, and the miniaturization thereofcan be achieved.

Application Example 19

According to this application example of the invention, there isprovided a surface acoustic wave oscillator including: theabove-described surface acoustic wave resonator; and an IC that is usedfor driving the IDT.

Accordingly, a surface acoustic wave oscillator having superioroscillation stability regardless of the use environment can be acquired.

Application Example 20

According to this application example of the invention, there isprovided an electronic apparatus including the above-described surfaceacoustic wave resonator.

Accordingly, an electronic apparatus having high reliability can beacquired.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described with reference to the accompanyingdrawings, wherein like numbers reference like elements.

FIGS. 1A to 1D are diagrams showing the configuration of a SAW deviceaccording to an embodiment. FIG. 1A is a diagram showing the planarconfiguration. FIG. 1B is a diagram showing a partially enlarged sectionof a side face. FIG. 1C is a partially enlarged view of FIG. 1B forillustrating details thereof. FIG. 1D is a partially enlarged view ofFIG. 1C and is a diagram showing a sectional shape of a groove portionthat can be assumed in a case where a SAW resonator is manufacturedusing a photolithography technique and an etching technique.

FIG. 2 is a diagram showing the orientation of a wafer that is a basematerial of a quartz crystal substrate used in an embodiment of theinvention.

FIGS. 3A and 3B are diagrams showing configuration examples of a SAWdevice in a case where a tilted type IDT is employed. FIG. 3A is anexample of a form in which electrode fingers are tilted so as to beperpendicular to the X′ axis. FIG. 3B is an example of a SAW devicehaving an IDT of which bus bars connecting electrode fingers are tilted.

FIG. 4 is a diagram showing a relationship between a stop band upper endmode and a stop band lower end mode.

FIG. 5 is a graph showing a relationship between the depth of aninter-electrode finger groove and a frequency variation amount in anoperating temperature range.

FIG. 6 is a diagram showing temperature characteristics in an ST cutquartz crystal substrate.

FIGS. 7A to 7D are graphs showing differences in changes of asecond-order temperature coefficient accompanying with a change in aline occupancy ratio η at a resonance point of the stop band upper endmode and a resonance point of the stop band lower end mode. FIG. 7A is agraph showing a displacement of the second-order temperature coefficientβ of the stop band upper end mode in a case where the depth of a grooveis 2% λ. FIG. 7B is a graph showing a displacement of the second-ordertemperature coefficient β of the stop band lower end mode in a casewhere the depth of a groove is 2% λ. FIG. 7C is a graph showing adisplacement of the second-order temperature coefficient β of the stopband upper end mode in a case where the depth of a groove is 4% λ. FIG.7D is a graph showing a displacement of the second-order temperaturecoefficient β of the stop band lower end mode in a case where the depthof a groove is 4% λ.

FIGS. 8A to 8I are graphs showing the relationship between the lineoccupancy ratio and the second-order temperature coefficient in a casewhere the depth of an inter-electrode finger groove is changed with thefilm thickness of the electrode set as zero. FIG. 8A is a graph in acase where the depth of the groove is 1% λ. FIG. 8B is a graph in a casewhere the depth of the groove is 1.25% λ. FIG. 8C is a graph in a casewhere the depth of the groove is 1.5% λ. FIG. 8D is a graph in a casewhere the depth of the groove is 2% λ. FIG. 8E is a graph in a casewhere the depth of the groove is 3% λ. FIG. 8F is a graph in a casewhere the depth of the groove is 4% λ. FIG. 8G is a graph in a casewhere the depth of the groove is 5% λ. FIG. 8H is a graph in a casewhere the depth of the groove is 6% λ. FIG. 8I is a graph in a casewhere the depth of the groove is 8% λ.

FIG. 9 is a graph showing the relationship between the depth of theinter-electrode finger groove at which the second-order temperaturecoefficient becomes zero and the line occupancy ratio η in a case wherethe film thickness of the electrode is set as zero.

FIGS. 10A to 10I are graphs showing the relationship between the lineoccupancy ratio and the amount of variation in the frequency in a casewhere the depth of the inter-electrode finger groove is changed with thefilm thickness of the electrode set as zero. FIG. 10A is a graph in acase where the depth of the groove is 1% λ. FIG. 10B is a graph in acase where the depth of the groove is 1.25% λ. FIG. 10C is a graph in acase where the depth of the groove is 1.5% λ. FIG. 10D is a graph in acase where the depth of the groove is 2% λ. FIG. 10E is a graph in acase where the depth of the groove is 3% λ. FIG. 10F is a graph in acase where the depth of the groove is 4% λ. FIG. 10G is a graph in acase where the depth of the groove is 5% λ. FIG. 10H is a graph in acase where the depth of the groove is 6% λ. FIG. 10I is a graph in acase where the depth of the groove is 8% λ.

FIG. 11 is a graph showing the relationship between the depth of theinter-electrode finger groove and the amount of variation in thefrequency in a case where the depth of the inter-electrode finger grooveis mismatched by ±0.001λ.

FIGS. 12A to 12F are graphs showing the relationship between the depthof the inter-electrode finger groove at which the second-ordertemperature coefficient is zero and the line occupancy ratio in a casewhere the film thickness of the electrode is changed. FIG. 12A is agraph in a case where the film thickness of the electrode is 1% λ. FIG.12B is a graph in a case where the film thickness of the electrode is1.5% λ. FIG. 12C is a graph in a case where the film thickness of theelectrode is 2% λ. FIG. 12D is a graph in a case where the filmthickness of the electrode is 2.5% λ. FIG. 12E is a graph in a casewhere the film thickness of the electrode is 3% λ. FIG. 12F is a graphin a case where the film thickness of the electrode is 3.5% λ.

FIGS. 13A and 13B are diagrams in which the relationship between η1 atwhich the second-order temperature coefficient is approximately zero(ppm/° C.²) for each film thickness of the electrode and the depth ofthe inter-electrode finger groove is organized in graphs. FIG. 13A showsthe relationship between the depth of the groove and η1 when the filmthickness of the electrode is changed from 1% λ to 3.5% λ. FIG. 13B is adiagram illustrating that an area for which |β|≦0.01 (ppm/° C.²) isinside a polygon formed by joining points a to h.

FIG. 14 is a diagram showing the relationship between the depth of theinter-electrode finger groove and the line occupancy ratio for the filmthickness of the electrode H≈0 to H=0.030λ as an approximate curve.

FIGS. 15A to 15F are graphs showing the relationship between the lineoccupancy ratio and the second-order temperature coefficient in a casewhere the depth of the inter-electrode finger groove is changed with thefilm thickness of the electrode set to 0.01λ. FIG. 15A is a graph in acase where the depth of the groove is 0. FIG. 15B is a graph in a casewhere the depth of the groove is 1% λ. FIG. 15C is a graph in a casewhere the depth of the groove is 2% λ. FIG. 15D is a graph in a casewhere the depth of the groove is 3% λ. FIG. 15E is a graph in a casewhere the depth of the groove is 4% λ. FIG. 15F is a graph in a casewhere the depth of the groove is 5% λ.

FIGS. 16A to 16F are graphs showing the relationship between the lineoccupancy ratio and the second-order temperature coefficient in a casewhere the depth of the inter-electrode finger groove is changed with thefilm thickness of the electrode set to 0.015λ. FIG. 16A is a graph in acase where the depth of the groove is 0. FIG. 16B is a graph in a casewhere the depth of the groove is 1% λ. FIG. 16C is a graph in a casewhere the depth of the groove is 1.5% λ. FIG. 16D is a graph in a casewhere the depth of the groove is 2.5% λ. FIG. 16E is a graph in a casewhere the depth of the groove is 3.5% λ. FIG. 16F is a graph in a casewhere the depth of the groove is 4.5% λ.

FIGS. 17A to 17F are graphs showing the relationship between the lineoccupancy ratio and the second-order temperature coefficient in a casewhere the depth of the inter-electrode finger groove is changed with thefilm thickness of the electrode set to 0.02λ. FIG. 17A is a graph in acase where the depth of the groove is 0. FIG. 17B is a graph in a casewhere the depth of the groove is 1% λ. FIG. 17C is a graph in a casewhere the depth of the groove is 2% λ. FIG. 17D is a graph in a casewhere the depth of the groove is 3% λ. FIG. 17E is a graph in a casewhere the depth of the groove is 4% λ. FIG. 17F is a graph in a casewhere the depth of the groove is 5% λ.

FIGS. 18A to 18F are graphs showing the relationship between the lineoccupancy ratio and the second-order temperature coefficient in a casewhere the depth of the inter-electrode finger groove is changed with thefilm thickness of the electrode set to 0.025λ. FIG. 18A is a graph in acase where the depth of the groove is 0. FIG. 18B is a graph in a casewhere the depth of the groove is 1% λ. FIG. 18C is a graph in a casewhere the depth of the groove is 1.5% λ. FIG. 18D is a graph in a casewhere the depth of the groove is 2.5% λ. FIG. 18E is a graph in a casewhere the depth of the groove is 3.5% λ. FIG. 18F is a graph in a casewhere the depth of the groove is 4.5% λ.

FIGS. 19A to 19F are graphs showing the relationship between the lineoccupancy ratio and the second-order temperature coefficient in a casewhere the depth of the inter-electrode finger groove is changed with thefilm thickness of the electrode set to 0.03λ. FIG. 19A is a graph in acase where the depth of the groove is 0. FIG. 19B is a graph in a casewhere the depth of the groove is 1% λ. FIG. 19C is a graph in a casewhere the depth of the groove is 2% λ. FIG. 19D is a graph in a casewhere the depth of the groove is 3% λ. FIG. 19E is a graph in a casewhere the depth of the groove is 4% λ. FIG. 19F is a graph in a casewhere the depth of the groove is 5% λ.

FIGS. 20A to 20F are graphs showing the relationship between the lineoccupancy ratio and the second-order temperature coefficient in a casewhere the depth of the inter-electrode finger groove is changed with thefilm thickness of the electrode set to 0.035λ. FIG. 20A is a graph in acase where the depth of the groove is 0. FIG. 20B is a graph in a casewhere the depth of the groove is 1% λ. FIG. 20C is a graph in a casewhere the depth of the groove is 2% λ. FIG. 20D is a graph in a casewhere the depth of the groove is 3% λ. FIG. 20E is a graph in a casewhere the depth of the groove is 4% λ. FIG. 20F is a graph in a casewhere the depth of the groove is 5% λ.

FIGS. 21A to 21F are graphs showing the relationship between the lineoccupancy ratio and the frequency variation amount in a case where thedepth of the inter-electrode finger groove is changed with the filmthickness of the electrode set to 0.01λ. FIG. 21A is a graph in a casewhere the depth of the groove is 0. FIG. 21B is a graph in a case wherethe depth of the groove is 1% λ. FIG. 21C is a graph in a case where thedepth of the groove is 2% λ. FIG. 21D is a graph in a case where thedepth of the groove is 3% λ. FIG. 21E is a graph in a case where thedepth of the groove is 4% λ. FIG. 21F is a graph in a case where thedepth of the groove is 5% λ.

FIGS. 22A to 22F are graphs showing the relationship between the lineoccupancy ratio and the frequency variation amount in a case where thedepth of the inter-electrode finger groove is changed with the filmthickness of the electrode set to 0.015λ. FIG. 22A is a graph in a casewhere the depth of the groove is 0. FIG. 22B is a graph in a case wherethe depth of the groove is 1% λ. FIG. 22C is a graph in a case where thedepth of the groove is 1.5% λ. FIG. 22D is a graph in a case where thedepth of the groove is 2.5% λ. FIG. 22E is a graph in a case where thedepth of the groove is 3.5% λ. FIG. 22F is a graph in a case where thedepth of the groove is 4.5% λ.

FIGS. 23A to 23F are graphs showing the relationship between the lineoccupancy ratio and the frequency variation amount in a case where thedepth of the inter-electrode finger groove is changed with the filmthickness of the electrode set to 0.02λ. FIG. 23A is a graph in a casewhere the depth of the groove is 0. FIG. 23B is a graph in a case wherethe depth of the groove is 1% λ. FIG. 23C is a graph in a case where thedepth of the groove is 2% λ. FIG. 23D is a graph in a case where thedepth of the groove is 3% λ. FIG. 23E is a graph in a case where thedepth of the groove is 4% λ. FIG. 23F is a graph in a case where thedepth of the groove is 5% λ.

FIGS. 24A to 24F are graphs showing the relationship between the lineoccupancy ratio and the frequency variation amount in a case where thedepth of the inter-electrode finger groove is changed with the filmthickness of the electrode set to 0.025λ. FIG. 24A is a graph in a casewhere the depth of the groove is 0. FIG. 24B is a graph in a case wherethe depth of the groove is 1% λ. FIG. 24C is a graph in a case where thedepth of the groove is 1.5% λ. FIG. 24D is a graph in a case where thedepth of the groove is 2.5% λ. FIG. 24E is a graph in a case where thedepth of the groove is 3.5% λ. FIG. 24F is a graph in a case where thedepth of the groove is 4.5% λ.

FIGS. 25A to 25F are graphs showing the relationship between the lineoccupancy ratio and the frequency variation amount in a case where thedepth of the inter-electrode finger groove is changed with the filmthickness of the electrode set to 0.03λ. FIG. 25A is a graph in a casewhere the depth of the groove is 0. FIG. 25B is a graph in a case wherethe depth of the groove is 1% λ. FIG. 25C is a graph in a case where thedepth of the groove is 2% λ. FIG. 25D is a graph in a case where thedepth of the groove is 3% λ. FIG. 25E is a graph in a case where thedepth of the groove is 4% λ. FIG. 25F is a graph in a case where thedepth of the groove is 5% λ.

FIGS. 26A to 26F are graphs showing the relationship between the lineoccupancy ratio and the frequency variation amount in a case where thedepth of the inter-electrode finger groove is changed with the filmthickness of the electrode set to 0.035λ. FIG. 26A is a graph in a casewhere the depth of the groove is 0. FIG. 26B is a graph in a case wherethe depth of the groove is 1% λ. FIG. 26C is a graph in a case where thedepth of the groove is 2% λ. FIG. 26D is a graph in a case where thedepth of the groove is 3% λ. FIG. 26E is a graph in a case where thedepth of the groove is 4% λ. FIG. 26F is a graph in a case where thedepth of the groove is 5% λ.

FIGS. 27A and 27B are diagrams showing the ranges in which |β|≦0.01based on the graph showing the relationship between the line occupancyratio and the depth of the groove in a case where the film thickness ofthe electrode is in the range of 0≦H<0.005λ. FIG. 27A is a diagram inthe case of η1, and FIG. 27B is a diagram in the case of η2.

FIGS. 28A and 28B are diagrams showing the ranges in which |β|≦0.01based on the graph showing the relationship between the line occupancyratio and the depth of the groove in a case where the film thickness ofthe electrode is in the range of 0.005λ≦H<0.010λ. FIG. 28A is a diagramin the case of η1, and FIG. 28B is a diagram in the case of η2.

FIGS. 29A and 29B are diagrams showing the ranges in which |β|≦0.01based on the graph showing the relationship between the line occupancyratio and the depth of the groove in a case where the film thickness ofthe electrode is in the range of 0.010λ≦H<0.015λ. FIG. 29A is a diagramin the case of η1, and FIG. 29B is a diagram in the case of η2.

FIGS. 30A and 30B are diagrams showing the ranges in which |β|≦0.01based on the graph showing the relationship between the line occupancyratio and the depth of the groove in a case where the film thickness ofthe electrode is in the range of 0.015λ≦H<0.020λ. FIG. 30A is a diagramin the case of η1, and FIG. 30B is a diagram in the case of η2.

FIGS. 31A and 31B are diagrams showing the ranges in which |β|≦0.01based on the graph showing the relationship between the line occupancyratio and the depth of the groove in a case where the film thickness ofthe electrode is in the range of 0.020λ≦H<0.025λ. FIG. 31A is a diagramin the case of η1, and FIG. 31B is a diagram in the case of η2.

FIGS. 32A and 32B are diagrams showing the ranges in which |β|≦0.01based on the graph showing the relationship between the line occupancyratio and the depth of the groove in a case where the film thickness ofthe electrode is in the range of 0.025λ≦H<0.030λ. FIG. 32A is a diagramin the case of η1, and FIG. 32B is a diagram in the case of η2.

FIGS. 33A and 33B are diagrams showing the ranges in which |β|≦0.01based on the graph showing the relationship between the line occupancyratio and the depth of the groove in a case where the film thickness ofthe electrode is in the range of 0.030λ≦H<0.035λ. FIG. 33A is a diagramin the case of η1, and FIG. 33B is a diagram in the case of η2.

FIGS. 34A to 34F are graphs showing the relationship between the depthof the inter-electrode finger groove and the Euler angle ψ when the filmthickness of the electrode and the line occupancy ratio (η1: solid line,η2: broken line) are determined. FIG. 34A is a graph in a case where thedepth of the groove is 1% λ. FIG. 34B is a graph in a case where thedepth of the groove is 1.5% λ. FIG. 34C is a graph in a case where thedepth of the groove is 2% λ. FIG. 34D is a graph in a case where thedepth of the groove is 2.5% λ. FIG. 34E is a graph in a case where thedepth of the groove is 3% λ. FIG. 34F is a graph in a case where thedepth of the groove is 3.5% λ.

FIG. 35 is a diagram in which the relationship between the depth of theinter-electrode finger groove and the Euler angle ψ for each filmthickness of the electrode is organized in a graph.

FIG. 36 is a graph showing the relationship between the depth of theinter-electrode finger groove at which the second-order temperaturecoefficient is −0.01 (ppm/° C.²) and the Euler angle ψ.

FIG. 37 is a graph showing the relationship between the depth of theinter-electrode finger groove at which the second-order temperaturecoefficient is +0.01 (ppm/° C.²) and the Euler angle ψ.

FIGS. 38A and 38B are graphs showing the ranges of ψ satisfying thecondition of |β|≦0.01 (ppm/° C.²) in a case where the film thickness ofthe electrode is in the range of 0<H≦0.005λ. FIG. 38A is a diagramrepresenting a maximum value and a minimum value of ψ, and FIG. 38B is adiagram representing the area of ψ satisfying the condition of β.

FIGS. 39A and 39B are graphs showing the ranges of ψ satisfying thecondition of |β|≦0.01 (ppm/° C.²) in a case where the film thickness ofthe electrode is in the range of 0.005λ<H≦0.010λ. FIG. 39A is a diagramrepresenting a maximum value and a minimum value of ψ, and FIG. 39B is adiagram representing the area of ψ satisfying the condition of β.

FIGS. 40A and 40B are graphs showing the ranges of ψ satisfying thecondition of |β|≦0.01 (ppm/° C.²) in a case where the film thickness ofthe electrode is in the range of 0.010λ<H≦0.015λ. FIG. 40A is a diagramrepresenting a maximum value and a minimum value of ψ, and FIG. 40B is adiagram representing the area of ψ satisfying the condition of β.

FIGS. 41A and 41B are graphs showing the ranges of ψ satisfying thecondition of |β|≦0.01 (ppm/° C.²) in a case where the film thickness ofthe electrode is in the range of 0.015λ<H≦0.020λ. FIG. 41A is a diagramrepresenting a maximum value and a minimum value of ψ, and FIG. 41B is adiagram representing the area of ψ satisfying the condition of β.

FIGS. 42A and 42B are graphs showing the ranges of ψ satisfying thecondition of |β|≦0.01 (ppm/° C.²) in a case where the film thickness ofthe electrode is in the range of 0.020λ<H≦0.025λ. FIG. 42A is a diagramrepresenting a maximum value and a minimum value of ψ, and FIG. 42B is adiagram representing the area of ψ satisfying the condition of β.

FIGS. 43A and 43B are graphs showing the ranges of ψ satisfying thecondition of |β|≦0.01 (ppm/° C.²) in a case where the film thickness ofthe electrode is in the range of 0.025λ<H≦0.030λ. FIG. 43A is a diagramrepresenting a maximum value and a minimum value of ψ, and FIG. 43B is adiagram representing the area of ψ satisfying the condition of β.

FIGS. 44A and 44B are graphs showing the ranges of ψ satisfying thecondition of |β|≦0.01 (ppm/° C.²) in a case where the film thickness ofthe electrode is in the range of 0.030λ<H≦0.035λ. FIG. 44A is a diagramrepresenting a maximum value and a minimum value of ψ, and FIG. 44B is adiagram representing the area of ψ satisfying the condition of β.

FIG. 45 is a graph showing the relationship between an Euler angle θ andthe second-order temperature coefficient in a case where the filmthickness of the electrode is 0.02λ and the depth of the inter-electrodefinger groove is 0.04λ.

FIG. 46 is a graph showing the relationship between a Euler angle φ andthe second-order temperature coefficient.

FIG. 47 is a graph showing the relationship between the Euler angle θand the Euler angle ψ at which the frequency-temperature characteristicsare good.

FIG. 48 is a diagram showing an example of the frequency-temperaturecharacteristic data for four test pieces under the condition at whichthe frequency-temperature characteristics are the best.

FIG. 49 is a graph showing the relationship between a level difference,which is a sum of the depth of the inter-electrode finger groove and thefilm thickness of the electrode and a CI value.

FIG. 50 is a table showing an example of an equivalent circuit constantand static characteristics of a SAW resonator according to thisembodiment.

FIG. 51 is impedance curve data of a SAW resonator according to thisembodiment.

FIG. 52 is a graph for comparing the relationship between the leveldifference and the Q value of a general SAW resonator and therelationship between the level difference and the Q value of a SAWresonator according to this embodiment.

FIG. 53 is a diagram showing the SAW reflection characteristics of anIDT and a reflector.

FIG. 54 is a graph showing the relationship between the film thicknessof the electrode and the frequency variance in a heat-cycle test.

FIGS. 55A and 55B are diagrams showing the configurations of a SAWoscillator according to an embodiment of the invention.

FIGS. 56A and 56B are graphs showing the frequency-temperaturecharacteristics of a SAW resonator. FIG. 56A is a graph showing thefrequency-temperature characteristics of a SAW resonator disclosed inJP-A-2006-203408, and FIG. 56B is a graph showing the range of thefrequency-temperature characteristics within a practical operatingtemperature range.

FIG. 57 is a graph showing the change in the amount of variation in thefrequency within the operating range of a SAW resonator in which an IDTand a reflector are coated with alumina as a protection film.

FIG. 58 is a graph showing the change in the amount of variation in thefrequency within the operating range of a SAW resonator in which an IDTand a reflector are coated with SiO₂ as a protection film.

FIG. 59 is a diagram showing another configuration example of a SAWdevice according to an embodiment and is a diagram showing a partiallyenlarged cross-section.

FIG. 60 is a diagram showing further another configuration example of aSAW device according to an embodiment and is a diagram showing apartially enlarged cross-section.

FIG. 61A represents the effective film thickness of an electrode fingerof the IDT in the horizontal axis and the Q value in the vertical axisand is a graph representing the trend of change in the Q value when theeffective film thickness is changed. FIG. 61B represents the effectivefilm thickness in the horizontal axis and the electromechanical couplingcoefficient in the vertical axis and is a graph representing the trendof change in the electromechanical coupling coefficient when theeffective film thickness is changed.

FIG. 62 is a diagram illustrating a SAW device according to a secondembodiment and is a diagram showing a partially enlarged cross-section.

FIG. 63 is a diagram showing another configuration example of the SAWresonator shown in FIG. 62.

FIG. 64 is a diagram illustrating a SAW device according to a thirdembodiment and is a diagram showing a partially enlarged cross-section.

FIG. 65 is a diagram illustrating a SAW device according to a fourthembodiment and is a diagram showing a partially enlarged cross-section.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereafter, a surface acoustic wave resonator, a surface acoustic waveoscillator, and an electronic apparatus according to embodiments of theinvention will be described in detail with reference to the accompanyingdrawings.

Surface Acoustic Wave Resonator First Embodiment

First a surface acoustic wave (SAW) resonator according to a firstembodiment of the invention will be described with reference to FIGS. 1Ato 1D. Of FIGS. 1A to 1D, FIG. 1A is a plan view of the SAW resonator,FIG. 1B is a partially enlarged sectional view, FIG. 1C is a partiallyenlarged view of FIG. 1B for illustrating details thereof, and FIG. 1Dis a partially enlarged view of FIG. 1C, is a sectional shape that canbe assumed in a case where a SAW resonator according to an embodiment ofthe invention is manufactured using a photolithography technique and anetching technique, and is a diagram illustrating a method of specifyinga line occupancy ratio η of IDT electrode fingers in a case where thecross-sectional shape is not a rectangle but a trapezoid. It isappropriate that the line occupancy ratio η is the ratio of the width Lof a convex portion to a value (L+S) acquired by adding the width L ofthe convex portion and a width S of a groove 32 at a height of ½ of avalue (G+H) acquired by adding the depth (the height of a pedestal) G ofthe groove 32 and an electrode film thickness H from the bottom portionof the groove 32.

The SAW resonator 10 according to this embodiment is basicallyconfigured by a quartz crystal substrate 30, an IDT 12, and reflectors20.

FIG. 2 is a diagram showing the orientation of a wafer 1 that is a basematerial of the quartz crystal substrate 30 used in an embodiment of theinvention. In FIG. 2, the X axis is the electrical axis of the quartzcrystal, the Y axis is the mechanical axis of the quartz crystal, andthe Z axis is the optical axis of the quartz crystal. The wafer 1 has aface that is acquired by rotating a face 2 perpendicular to the Y axisabout the X axis as the rotation axis from the +Z axis side toward the−Y axis side by an angle of θ′ degrees. An axis perpendicular to therotated face is the Y′ axis, and an axis that is parallel to the rotatedface and is perpendicular to the X axis is the Z′ axis. Furthermore, theIDT 12 and the reflectors 20 configuring the SAW resonator 10 aredisposed along the X′ axis that is an axis acquired by rotating the Xaxis of the quartz crystal about the Y′ axis as the rotation axis by +ψdegrees (or −ψ degrees) with the rotation direction from the +X axisside toward the +Z′ axis defined as positive. The quartz crystalsubstrate 30 configuring the SAW resonator 10 is diced by being cut outof the wafer 1. Although the shape of the quartz crystal substrate 30 ina plan view is not particularly limited, it may, for example, be arectangle having short sides parallel to a Z″ axis that is an axisacquired by rotating the Z′ axis by +ψ degrees about the Y′ axis as therotation axis and long sides parallel to the X′ axis. Here, therelationship between θ′ and an Euler angle θ is θ′=θ−90°.

In this embodiment, an in-plane rotation ST cut quartz crystal substratethat is represented by Euler angles (−1.5°≦φ≦1.5°, 117°≦θ≦142°, and42.79°≦|ψ|≦49.57°) is employed as the quartz crystal substrate 30. Here,the Euler angles will be described. A substrate represented by Eulerangles (0°, 0°, 0°) is a Z cut substrate that has a main faceperpendicular to the Z axis. Here, φ of Euler angles (φ, θ, ψ) relatesto first rotation of the Z cut substrate and is a first rotation anglein which the Z axis is set as the rotation axis, and a direction forrotating from the +X axis side to the +Y axis side is defined as apositive rotation angle. In addition, the Euler angle θ relates tosecond rotation, which is performed after the first rotation of the Zcut substrate and is a second rotation angle in which the X axis afterthe first rotation is set as the rotation axis, and a direction forrotating from the +Y axis after the first rotation to the +Z axis isdefined as a positive rotation angle. The cut surface of a piezoelectricsubstrate is determined by the first rotation angle φ and the secondrotation angle θ. The Euler angle ψ relates to third rotation that isperformed after the second rotation of the Z cut substrate and is athird rotation angle in which the Z axis after the second rotation isset as the rotation axis, and a direction for rotating from the +X axisside after the second rotation to the +Y axis side after the secondrotation is defined as a positive rotation angle. A propagationdirection of the SAW is represented by the third rotation angle ψ withrespect to the X axis after the second rotation.

The IDT 12 includes a pair of inter digital transducers 14 a and 14 b ofwhich the base end portions of a plurality of electrode fingers 18 a and18 b are connected to bus bars 16 a and 16 b, and the electrode finger18 a configuring one inter digital transducer 14 a and the electrodefinger 18 b configuring the other inter digital transducer 14 b arealternately disposed with a predetermined gap maintained therebetween inthe IDT 12. In addition, the electrode fingers 18 a and 18 b shown inFIG. 1A are disposed in such a way that the extension direction of theelectrode fingers 18 a and 18 b is perpendicular to the X′ axis that isthe propagation direction of the surface acoustic wave. A SAW excited bythe SAW resonator 10 configured in such a way is a Rayleigh type SAW andhas oscillation displacement components for both the Y′ axis and X′axis. By displacing the propagation direction of the SAW from the Xaxis, which is the crystal axis of the quartz crystal as above, it ispossible to excite a SAW of the stop band upper end mode.

Furthermore, the SAW resonator 10 according to an embodiment of theinvention can be configured in a form as shown in FIGS. 3A and 3B. Inother words, even in the case where an IDT tilted by a power flow angle(hereafter, referred to as a PFA) δ from the X′ axis as shown in FIGS.3A and 3B is applied, a high Q can be implemented by satisfying thefollowing requirements. FIG. 3A is a plan view showing a tilted-type IDT12 a according to an embodiment. In the tilted-type IDT 12 a, thedisposition form of the electrode fingers 18 a and 18 b is tilted suchthat the X′ axis, which is the propagation direction of the SAWdetermined by the Euler angles, and the direction of the electrodefingers 18 a and 18 b of the tilted type IDT 12 a are perpendicular toeach other.

FIG. 3B is a plan view showing a tilted type IDT 12 a according toanother embodiment. In this example, although the direction of thearrangement of the electrode fingers is disposed so as to be tilted withrespect to the X′ axis by tilting the bus bars 16 a and 16 b thatconnect the electrode fingers 18 a and 18 b, the tilted IDT 12 a isconfigured such that the X′ axis and the extension direction of theelectrode fingers 18 a and 18 b are perpendicular to each other,similarly to the configuration shown in FIG. 3A.

No matter which tilted type IDT is used, by disposing the electrodefingers such that a direction perpendicular to the X′ axis is theextension direction of the electrode fingers as in these embodiments, itis possible to realize a low-loss SAW resonator with good temperaturecharacteristics according to an embodiment of the invention beingmaintained.

Here, the relationship between a SAW of the stop band upper end mode anda SAW of the stop band lower end mode will be described. The positionsof the anti-nodes (or nodes) of the standing waves of a SAW of the stopband lower end mode and a SAW of the stop band upper end mode that areformed by the normal IDT 12 as shown in FIG. 4 (the electrode fingers 18configuring the IDT 12 are shown in FIG. 4) are misaligned from eachother by π/2. FIG. 4 is a diagram showing the distribution of standingwaves of the stop band upper end mode and the stop band lower end modein the normal IDT 12.

As shown in FIG. 4, as described above, the anti-nodes of the standingwave of the stop band lower end mode that is denoted by a solid line arelocated at the center positions of the electrode fingers 18, that is,the center positions of reflection, and the nodes of the standing waveof the stop band upper end mode that is denoted by a dashed-dotted lineare located at the center positions of reflection. In such a mode inwhich the nodes are located at the center positions between theelectrode fingers, there are many cases where the oscillation of the SAWcannot be efficiently converted into electric charge by the electrodefingers 18 (18 a and 18 b) and the standing wave of the mode cannot beexcited or received as an electrical signal. However, according to thetechnique described here, by setting the Euler angle ψ to a non-zerovalue and displacing the propagation direction of the SAW from the Xaxis, which is the crystal axis of the quartz crystal, the standing waveof the stop band upper end mode can be shifted to the position of thesolid line shown in FIG. 4, in other words, the anti-nodes of thestanding wave of the mode can be shifted to the center positions of theelectrode fingers 18, whereby the SAW of the stop band upper end modecan be excited.

In addition, one pair of the reflectors (reflection units) 20 aredisposed such that the IDT 12 is interposed therebetween in thepropagation direction of the SAW. As a specific configuration example,both ends of each of a plurality of conductive strips 22 that aredisposed to be parallel to the electrode fingers 18 configuring the IDT12 are connected together.

As the material of the electrode film that configures the IDT 12 and thereflectors 20 configured as above, aluminum (Al), or an alloy with Alused as its main constituent can be used.

By decreasing the electrode thickness of the electrode film configuringthe IDT 12 and reflectors 20 as much as possible, the influence of thetemperature characteristics of the electrodes is minimized. Furthermore,by increasing the depth of the grooves of the quartz crystal substrateportion, good frequency-temperature characteristics are derived from theperformance of the grooves of the quartz crystal substrate portion, inother words, by utilizing the good temperature characteristics of thequartz crystal. Accordingly, the influence of the temperaturecharacteristics of the electrode on the temperature characteristics ofthe SAW resonator can be reduced, and, in a case where the variation ofthe mass of the electrode is within 10%, good temperaturecharacteristics can be maintained.

In a case where an alloy is used as the material of the electrode filmfor the above-described reasons, the ratio by weight of metals otherthan Al, which is the main component, is equal to or less than 10% andis preferably equal to or less than 3%. In a case where electrodes usinga metal other than Al as their main constituent are used, it ispreferable that the film thickness of the electrode is adjusted suchthat the mass of the electrode is within ±10% of that in a case where Alis used. In such a case, good temperature characteristics equivalent tothose of a case where Al is used can be obtained.

In the quartz crystal substrate 30 of the SAW resonator 10 having theabove-described basic configuration, a plurality of the grooves(inter-electrode finger grooves) 32 that are arranged so as to beparallel to the electrode fingers 18 is disposed between the electrodefingers of the IDT 12 or the conductive strips 22 of the reflectors 20.

When the wavelength of the SAW in the stop band upper end mode isdenoted by λ, and the depth of the groove is denoted by G, the groove 32arranged in the quartz crystal substrate 30 may satisfy the followingrelationship.0.01λ≦G  (1)

In addition, in a case where the upper limit of the depth G of thegroove is to be determined, as can be acquired by referring to FIG. 5,the depth G of the groove may be in the following range.0.01λ≦G≦0.094λ  (2)

The reason for this is that, by fixing the depth G of the groove withinsuch a range, the amount of frequency variation in the operatingtemperature range (−40° C. to +85° C.) can be controlled so as to beequal to or less than 25 ppm as a target value to be described later indetail. In addition, it is preferable that the depth G of the groove iswithin the following range.0.01λ≦G≦0.0695λ  (3)

By setting the depth G of the groove to be in such a range, even in acase where there is manufacturing variation in the depth G of thegroove, the amount of shift in the resonance frequency of an individualSAW resonator 10 can be suppressed within a correctable range.

In addition, the above-described wavelength λ of the SAW is thewavelength of the SAW near the IDT 12.

In addition, the line occupancy ratio η, as illustrated in FIGS. 1C and1D, is a value that is acquired by dividing the line width L of theelectrode finger 18 (the width of a convex portion in a case where theelectrode finger is configured only by the crystal quartz convexportion) by the pitch λ/2 (=L+S) between the electrode fingers 18.Accordingly, the line occupancy ratio η can be represented as thefollowing Equation (4).η=L/(L+S)  (4)

In the SAW resonator 10 according to this embodiment, the line occupancyratio η may be set in the range satisfying Equations (5) and (6). As canbe understood from Equations (5) and (6), the line occupancy ratio η canbe derived by determining the depth G of the groove 32.−2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775, wherein 0.0100λ≦G≦0.0500λ  (5)−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775, wherein 0.0500λ<G≦0.0695λ  (6)

In addition, the film thickness of the material of the electrode film(the IDT 12, the reflector 20, and the like) of the SAW resonator 10according to this embodiment is preferably in the following range.0<H≦0.035λ  (7)

Furthermore, in a case where the thickness of the electrode filmrepresented in Equation (7) is considered for the line occupancy ratioη, the line occupancy ratio η can be acquired by using Equation (8).η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)  (8)

The manufacturing variation in the electric characteristics(particularly, the resonance frequency) increases as the film thicknessof the electrode is increased. Accordingly, there is high possibilitythat the line occupancy ratio η has manufacturing variation within ±0.04in a case where the film thickness H of the electrode is in the rangerepresented in Equations (5) and (6) and manufacturing variation largerthan ±0.04 when H>0.035λ. However, in a case where the film thickness Hof the electrode is within the range represented in Equations (5) and(6), and the variation of the line occupancy ratio η is within ±0.04, aSAW device having a small second-order temperature coefficient β can berealized. In other words, the line occupancy ratio η can be allowed tobe in a range represented in Equation (9) that is acquired by adding acommon difference of ±0.04 to Equation (8).η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)±0.04  (9)

In the SAW resonator 10 having the above-described configurationaccording to this embodiment, the object is to improve thefrequency-temperature characteristics up to the degree at which theamount ΔF of change in the frequency is equal to or less than 25 ppm inan operating temperature in a case where the second-order temperaturecoefficient β is within ±0.01 (ppm/° C.²) and the desired operatingtemperature range of the SAW is −40° to +85° C.

However, generally, the temperature characteristics of the surfaceacoustic wave resonator are represented in the following equation.Δf=α×(T−T ₀)+β×(T−T ₀)²

Here, Δf represents the amount (ppm) of change in the frequency betweenthe temperature T and the apex temperature T₀, α represents thefirst-order temperature coefficient (ppm/° C.), β represents thesecond-order temperature coefficient (ppm/° C.²), T representstemperature, and T₀ represents the temperature (apex temperature) atwhich the frequency is the maximum.

For example, in a case where the piezoelectric substrate is configuredby a quartz crystal substrate of so-called ST-cut (Euler angles (φ, θ,ψ)=(0°, 120° to 130°, 0°)), the first-order temperature coefficientα=0.0, and the second-order temperature coefficient β=−0.034, and agraph as shown in FIG. 6 is formed. In FIG. 6, the temperaturecharacteristics are drawn as an upward-convex parabola (second-ordercurve).

In the SAW resonator as shown in FIG. 6, since the amount of variance inthe frequency with respect to a change in the temperature is extremelylarge, the amount Δf of change in the frequency with respect to thechange in the temperature needs to be suppressed. Accordingly, a surfaceacoustic wave resonator needs to be implemented based on a new conceptsuch that the amount Δf of change in the frequency with respect to thechange in the temperature (operating temperature) at a time when the SAWresonator is actually used is close to zero by allowing the second-ordertemperature coefficient β shown in FIG. 6 to move closer to zero.

Thus, one of the objects of an embodiment of the invention is to solvethe above-described problems so as to realize a surface acoustic wavedevice operating at a stable frequency even in a case where thetemperature changes by improving the frequency-temperaturecharacteristics of the surface acoustic wave device to be extremelyexcellent.

Hereinafter, the implementation of a SAW device having the configurationin which the above-described technical idea (technical elements) iscontained for solving the above-described problems, that is, how theconcept according to an embodiment of the invention has been reached byrepeating simulation and experiments performed by inventors of thisapplication will be described and proved in detail.

In a SAW resonator in which the direction of propagation is set to thecrystal X-axis direction by using a quartz crystal substrate called theabove-described ST cut, in a case where an operating temperature rangeis the same, the amount ΔF of change in the frequency within theoperating temperature range is about 133 (ppm), and the second-ordertemperature coefficient β is about −0.034 (ppm/° C.²). In addition, in aSAW resonator using an in-plane rotated ST cut quartz crystal substratein which the cut angle of the quartz crystal substrate and thepropagation direction of the SAW are represented by Euler angles (0°,123°, 45°), and the operating temperature range is the same, the amountΔF of change in the frequency is about 63 ppm, and the second-ordertemperature coefficient β is about −0.016 (ppm/° C.²) in a case whereexcitation of the stop band lower end mode is used.

Since a SAW resonator that uses the ST-cut quartz crystal substrate orthe in-plane rotated ST-cut quartz crystal substrate uses a surfaceacoustic wave called Rayleigh wave, and the surface acoustic wave calledRayleigh wave has the variation in the frequency or thefrequency-temperature characteristics with respect to the processingprecision of the quartz crystal substrate or the quartz crystalsubstrate that is much smaller than a surface acoustic wave called aleaky wave of the LST-cut quartz crystal substrate, and accordingly, themass producibility thereof is superior, and the SAW resonator is used invarious SAW devices. However, a general SAW resonator using the ST-cutquartz crystal substrate, the in-plane rotated ST-cut quartz crystalsubstrate, or the like, as described above, has the second-ordertemperature characteristics in which a curve representing thefrequency-temperature characteristics is a second-order curve, andfurthermore, the absolute value of the second-order temperaturecoefficient of the second-order characteristics is large. Accordingly,such a SAW resonator cannot be used in a SAW device such as a resonator,an oscillator, or the like that is used in a wired communication deviceor a radio communication device in which the amount of frequencyvariation in the operating temperature range is large and the stabilityof the frequency is required. For example, in a case where thefrequency-temperature characteristics having the second-ordertemperature characteristics in which the second-order temperaturecoefficient β is equal to or less than ±0.1 (ppm/° C.²) corresponding tothe improvement equal to or less than ⅓ of the second-order temperaturecoefficient β of the ST-cut quartz crystal substrate or the improvementequal to or more than 37% of the second-order temperature coefficient βof the in-plane rotated ST-cut quartz crystal substrate is acquired, adevice requiring such frequency stability can be realized. Furthermore,in a case where third-order temperature characteristics are acquired inwhich the second-order temperature coefficient β is almost zero, and acurve representing the frequency-temperature characteristics is athird-order curve, the frequency stability is further improved for theoperating temperature range, which is more preferable. According to thethird-order temperature characteristics, a SAW device having the amountof change in the frequency that is equal to or less than ±25 ppm, whichcannot be realized by a general SAW device, is realized also for a broadoperating temperature range of −40° C. to +85° C., whereby extremelyhigh frequency stability is acquired.

It becomes apparent that the line occupancy ratio η of the electrodefingers 18 of the IDT 12, the film thickness H of the electrode, thedepth G of the groove, and the like relate to the change in thefrequency-temperature characteristics of the SAW resonator 10, asdescribed above, based on the findings that are based on the simulationsand experiments performed by the inventors of this application. The SAWresonator 10 according to this embodiment uses the excitation of thestop band upper end mode.

FIGS. 7A to 7D are graphs showing the changes in the second-ordertemperature coefficient β with respect to the change in the lineoccupancy ratio η in a case where a SAW is excited so as to propagate onthe surface of the quartz crystal substrate 30 in a state in which thefilm thickness H of the electrode is zero (H=0% λ) in FIG. 1C, that is,a state in which grooves 32 formed from concave-convex quartz crystalare formed on the surface of the quartz crystal substrate 30. AmongFIGS. 7A to 7D, FIG. 7A shows the second-order temperature coefficient βin the resonance of the stop band upper end mode in a case where thedepth G of the grooves is 0.02λ, and FIG. 7B shows the second-ordertemperature coefficient β in the resonance of the stop band lower endmode in a case where the depth G of the grooves is 0.02λ. In addition,among FIGS. 7A to 7D, FIG. 7C shows the second-order temperaturecoefficient β in the resonance of the stop band upper end mode in a casewhere the depth G of the grooves is 0.04λ, and FIG. 7D shows thesecond-order temperature coefficient β in the resonance of the stop bandlower end mode in a case where the depth G of the grooves is 0.04λ. Thesimulations illustrated in FIGS. 7A to 7D show examples in each casewhere a SAW propagates in a form on the quartz crystal substrate 30 inwhich the electrode film is not arranged so as to decrease the factorschanging the frequency-temperature characteristics. Here, as the cutangle of the quartz crystal substrate 30, an angle corresponding toEuler angles (0°, 123°, ψ) is used. In addition, the value of ψ isappropriately set for which the absolute value of the second-ordertemperature coefficient β is the minimum.

It can be read from FIGS. 7A to 7D that the second-order temperaturecoefficient β abruptly changes near the line occupancy ratio η of 0.6 to0.7 in the case of the stop band upper end mode or the stop band lowerend mode. Then, by comparing the change in the second-order temperaturecoefficient β in the stop band upper end mode and the change in thesecond-order temperature coefficient β in the stop band lower end mode,the following can be read out. The change in the second-ordertemperature coefficient β in the stop band lower end mode changes fromthe negative side to a further negative side, whereby thecharacteristics deteriorate (the absolute value of the second-ordertemperature coefficient β increases). In contrast to this, the change inthe second-order temperature coefficient β in the stop band upper endmode changes from the negative side to the positive side, whereby thecharacteristics are improved (there is a point at which the absolutevalue of the second-order temperature coefficient β decreases).

Accordingly, it is apparent that, in order to acquire goodfrequency-temperature characteristics of a SAW device, the oscillationof the stop band upper end mode is preferably used.

Next, the inventors checked the relationship between the line occupancyratio η and the second-order temperature coefficient β in a case where aSAW of the stop band upper end mode is allowed to propagate on a quartzcrystal substrate of which the depth G of the grooves is variouslychanged.

FIGS. 8A to 8I are graphs each showing the evaluation result ofsimulating the relationship between the line occupancy ratio η and thesecond-order temperature coefficient β in a case where, similarly toFIGS. 7A to 7D, the film thickness H of the electrode is set to zero(H=0% λ), and the depth G of the grooves is changed from 0.01λ (1% λ) to0.08λ (8% λ). It can be read from the evaluation results that a point atwhich β=0, that is, a point at which an approximate curve representingthe frequency-temperature characteristics represents a third-order curvestarts to appear from around a point at which the depth G of the groovesis 0.0125λ (1.25%) as represented in FIG. 8B. In addition, from FIGS. 8Ato 8I, it is determined that there are two positions of η (a point (η1)at which β=0 for a larger η and a point (η2) at which β=0 for a smallerη) at which β=0. In addition, it can be read from the evaluation resultsshown in FIGS. 8A to 8I that the amount of change in the line occupancyratio η with respect to the change in the depth G of the grooves islarger for η2 than η1.

The understanding of this point can be deepened by referring to FIG. 9.FIG. 9 is a graph in which η1 and η2, at which the second-ordertemperature coefficient β is zero, in a case where the depth G of thegrooves is changed, are plotted. It can be read from FIG. 9 that as thedepth G of the grooves is increased, η1 and η2 decrease, and the amountof change in η2 is large enough to scale out at around a point at whichthe depth G of the grooves is 0.04λ in the graph in which the scale ofthe vertical axis η is represented in the range of 0.5λ to 0.9λ. Inother words, it can be stated that the amount of change in η2 withrespect to the change in the depth G of the grooves is large.

FIGS. 10A to 10I are graphs that represent the amount ΔF of change inthe frequency by setting the film thickness H of the electrode to zero(H=0% λ), similarly to FIGS. 7A to 7D and FIGS. 8A to 8I, and convertingthe vertical axis of FIGS. 8A to 8I into the second-order temperaturecoefficient β. As is apparent from FIGS. 10A to 10I, it can be read thatthe amount ΔF of change in the frequency decreases at two points (η1 andη2) at which β=0. Furthermore, from FIG. 10A to 10I, it can be read thatthe amount ΔF of change in the frequency at a point corresponding to η1out of the two points at which β=0 is suppressed less also in any graphin which the depth G of the grooves is changed.

According to the above-described tendency, for products manufacturedthrough mass production for which an error can easily occur at the timeof manufacturing, it is preferable to employ a point, at which β=0, ofwhich the amount of change in the frequency with respect to the variancein the depth G of the grooves is smaller, that is, η1. FIG. 5 shows agraph representing the relationship between the amount ΔF of change inthe frequency and the depth G of the grooves at a point (η1) at whichthe second-order temperature coefficient β is the minimum for each depthG of the grooves. From FIG. 5, the lower limit value of the depth G ofthe grooves at which the amount ΔF of change in the frequency is equalto or less than 25 ppm as a target value is 0.01λ, and the range of thedepth G of the grooves is equal to or more than the lower limit value,in other words, 0.01λ≦G.

In addition, examples are also added to FIG. 5 in which the depth G ofthe grooves is equal to or larger than 0.08λ through simulation.According to this simulation, the amount ΔF of change in the frequencyis equal to or less than 25 ppm in a case where the depth G of thegrooves is equal to or larger than 0.01λ, and thereafter, as the depth Gof the grooves increases, the amount ΔF of change in the frequencydecreases. However, in a case where the depth G of the grooves is equalto or larger than 0.09λ, the amount ΔF of change in the frequencyincreases again, and in a case where the depth G of the grooves exceeds0.094λ, the amount ΔF of change in the frequency exceeds 25 ppm.

Although the graph shown in FIG. 5 is the simulation in a state in whichthe electrode film such as the IDT 12 or the reflector 20 is not formedon the quartz crystal substrate 30, as can be understood by referring toFIGS. 21A to 26F that will be represented in detail later, it is thoughtthat the amount ΔF of change in the frequency decreases in a case wherethe electrode film is arranged in the SAW resonator 10. Accordingly,when the upper limit value of the depth G of the grooves is to bedetermined, it may be a maximum value in the state in which theelectrode film is not formed, that is, G≦0.094λ, and a preferred rangeof the depth G of the grooves for achieving the target can berepresented in the following equation.0.01λ≦G≦0.094λ  (2)

In addition, the depth G of the grooves has variations about a maximum±0.001λ in the mass production process. Accordingly, in a case where theline occupancy ratio η is set to be constant, the amount Δf of change inthe frequency of the SAW resonator 10 in a case where the depth G of thegrooves is deviated by ±0.001λ is shown in FIG. 11. As shown in FIG. 11,for the case of G=0.04λ, in a case where the depth G of the grooves isdeviated by ±0.001λ, in other words, in a case where the depth of thegrooves is in the range of 0.039λ≦G≦0.041λ, it can be read that theamount Δf of change in the frequency is about ±500 ppm.

Here, in a case where the amount Δf of change in the frequency is lessthan ±1000 ppm, frequency adjustment can be performed by using variousfine-frequency adjustment unit. However, in a case where the amount Δfof change in the frequency is equal to or larger than ±1000 ppm, staticcharacteristics such as a Q value and a CI (crystal impedance) value andlong-term reliability are affected by the frequency adjustment, whichleads to a decrease in the good product rate of the SAW resonator 10.

When an approximate equation that represents the relationship betweenthe amount Δf [ppm] of change in the frequency and the depth G of thegrooves is derived from a straight line connecting the plots shown inFIG. 11, Equation (10) can be acquired.Δf=16334(G/λ)−137  (10)

Here, when the value of G at which Δf<1000 ppm is acquired, G≦0.0695λ.Accordingly, a preferred range of the depth G of the grooves accordingto this embodiment is preferably a range as shown in Equation (3).0.01λ≦G≦0.0695λ  (3)

Next, FIGS. 12A to 12F show the graphs of evaluation results when therelationship between η at which the second-order temperature coefficientβ=0, that is, a line occupancy ratio η representing the third-ordertemperature characteristics and the depth G of the grooves. The Eulerangles of the quartz crystal substrate 30 are set to (0°, 123°, ψ).Here, as ψ, an angle at which the frequency-temperature characteristicsrepresent the tendency of a third-order curve, that is, an angle atwhich the second-order temperature coefficient β=0 is appropriatelyselected. In addition, under the conditions as shown in FIGS. 12A to12F, the relationship between the Euler angle ψ and the depth G of thegrooves when η at which β=0 is acquired is shown in FIGS. 34A to 34F. Ina graph (FIG. 34C) for the film thickness H of the electrode=0.02λ,although a plot for ψ<42° is not represented, in the plot of this graphfor η2, ψ=41.9° at G=0.03λ. The plot for the relationship between thedepth G of the grooves and the line occupancy ratio η for each filmthickness of the electrode is acquired based on FIGS. 15A to 20F thatwill be described later in detail.

From the evaluation results shown in FIGS. 12A to 12F, for any filmthickness, it can be read that the variation with respect to the changein the depth G of the grooves at η1 is smaller than that at η2 asdescribed above. Accordingly, η1 is extracted from the graphrepresenting the relationship between the depth G of the grooves and theline occupancy ratio η for each film thickness shown in FIGS. 12A to12F, and points at which β≈0 are collectively plotted in FIG. 13A.Meanwhile an area at which β≈0 is not satisfied but |β|≦0.01 issatisfied is evaluated, and it is apparent that η1 is concentratedwithin a polygon denoted by solid lines as shown in FIG. 13B.

The coordinates of points a to h shown in FIG. 13B are represented inthe following Table 1.

TABLE 1 Point G/λ η a 0.01 0.70 b 0.03 0.66 c 0.05 0.62 d 0.07 0.55 e0.07 0.60 f 0.05 0.65 g 0.03 0.70 h 0.01 0.75

FIG. 13B represents that |β|≦0.01 is assured within a polygon surroundedby points a to h regardless of the thickness of the film thickness H ofthe electrode, and good frequency-temperature characteristics can beacquired therein. The range in which the good frequency-temperaturecharacteristics can be acquired is a range that satisfies both Equations(11), (12), and (13) shown below.η≦−2.5000×G/λ+0.7775, wherein 0.0100λ≦G≦0.0695λ  (11)η≧−2.0000×G/λ+0.7200, wherein 0.0100λ≦G≦0.0500λ  (12)η≧−3.5898×G/λ+0.7995, wherein 0.0500λ<G≦0.0695λ  (13)

Based on Equations (11), (12), and (13), it can be stated that, for therange surrounded by the solid lines shown in FIG. 13B, the lineoccupancy ratio η can be specified as a range that satisfies bothEquations (5) and (6).−2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775, wherein 0.0100λ≦G≦0.0500λ  (5)−3.5898×G/λ+0.7995≦η≦2.5000×G/λ+0.7775, wherein 0.0500λ<G≦0.0695λ  (6)

Here, in a case where the second-order temperature coefficient β isallowed within ±0.01 (ppm/° C.²), it is checked that the second-ordertemperature coefficient β is within ±0.01 (ppm/° C.²) by configuringsuch that both Equations (3) and (5) are satisfied for 0.0100λ≦G≦0.0500λand both Equations (3) and (6) are satisfied for 0.0500λ≦G≦0.0695λ.

In addition, the values of the second-order temperature coefficients βof the film thickness H of each electrode at points a to h are shown inthe following Table 2. Based on Table 2, it can be checked that |β|≦0.01for all the points.

TABLE 2 Film Thickness H of Electrode Point 1% λ 1.5% λ 2% λ 2.5% λ 3% λ3.5% λ a −0.0099 −0.0070 −0.0030 0.0030 −0.0050 −0.0060 b 0.0040 0.00300.0000 0.0000 −0.0020 −0.0040 c 0.0070 −0.0040 0.0010 −0.0036 −0.0040−0.0057 d 0.0067 −0.0022 −0.0070 −0.0080 −0.0090 −0.0099 e −0.0039−0.0060 −0.0090 −0.0080 −0.0090 −0.0094 f −0.0023 −0.0070 −0.0050−0.0062 −0.0060 −0.0070 g −0.0070 −0.0060 −0.0090 −0.0070 −0.0070−0.0070 h −0.0099 −0.0030 −0.0091 −0.0080 −0.0080 −0.0080

When the relationship between the depth G of the grooves at which β=0and the line occupancy ratio η of the SAW resonator 10 in which the filmthickness of the electrode H≈0, 0.01λ, 0.02λ, 0.03λ, or 0.035λ isrepresented as an approximate line based on Equations (11) to (13) andEquations (5) and (6) derived therefrom, graphs shown in FIG. 14 areformed. In addition, the relationship between the depth G of the groovesand the line occupancy ratio η of the quartz crystal substrate 30 inwhich the electrode film is not arranged is as shown in FIG. 9.

When the film thickness H of the electrode is changed to be equal to orless than 3.0% λ(0.030λ), β=0, that is, the frequency-temperaturecharacteristics of a third-order curve can be acquired. At this time,the relationship between G and η at which the frequency-temperaturecharacteristics are good can be represented as in Equation (8).η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)  (8)

Here, the unit of G and H is λ.

However, this Equation (8) satisfies for the film thickness H of theelectrode that is in the range of 0<H≦0.030λ.

The manufacturing variation in the electric characteristics(particularly, the resonance frequency) increases as the film thicknessof the electrode is increased. Accordingly, there is high possibilitythat the line occupancy ratio η has manufacturing variation that islarger than ±0.04 for the manufacturing variation within ±0.04 andH>0.035λ in the case where the film thickness H of the electrode is inthe range represented in Equations (5) and (6). However, in a case wherethe film thickness H of the electrode is within the range represented inEquations (5) and (6), and the variation of the line occupancy ratio ηis within ±0.04, a SAW device having a small second-order temperaturecoefficient β can be realized. In other words, in a case where, afterthe manufacturing variation of the line occupancy ratio is considered,the second-order temperature coefficient β is set within ±0.01 ppm/°C.², the line occupancy ratio η can be allowed to be in a rangerepresented in Equation (9) that is acquired by adding a commondifference of ±0.04 to Equation (8).η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)±0.04  (9)

FIGS. 15A to 20F show graphs illustrating the relationship between theline occupancy ratio η and the second-order temperature coefficient β ina case where the depth G of the grooves is changed for cases where thefilm thicknesses of the electrode are 0.01λ (1% λ), 0.015λ (1.5% λ),0.02λ (2% λ), 0.025λ (2.5% λ), 0.03λ (3% λ), and 0.035λ (3.5% λ).

In addition, FIGS. 21A to 26F show graphs illustrating the relationshipbetween the line occupancy ratio η and the amount ΔF of change in thefrequency in SAW resonators 10 corresponding to FIGS. 15A to 20F.Furthermore, the quartz crystal substrate having the Euler angles of(0°, 123°, ψ) is used, and an angle is appropriately selected as ψ suchthat ΔF is the minimum.

Here, FIGS. 15A to 15F are diagrams illustrating the relationshipbetween the line occupancy ratio η and the second-order temperaturecoefficient β in a case where the film thickness H of the electrode isset to 0.01λ, and FIGS. 21A to 21F are diagrams illustrating therelationship between the line occupancy ratio η and the amount ΔF ofchange in the frequency in a case where the film thickness H of theelectrode is set to 0.01λ.

Here, FIGS. 16A to 16F are diagrams illustrating the relationshipbetween the line occupancy ratio η and the second-order temperaturecoefficient β in a case where the film thickness H of the electrode isset to 0.015λ, and FIGS. 22A to 22F are diagrams illustrating therelationship between the line occupancy ratio η and the amount ΔF ofchange in the frequency in a case where the film thickness H of theelectrode is set to 0.015λ.

Here, FIGS. 17A to 17F are diagrams illustrating the relationshipbetween the line occupancy ratio η and the second-order temperaturecoefficient β in a case where the film thickness H of the electrode isset to 0.02λ, and FIGS. 23A to 23F are diagrams illustrating therelationship between the line occupancy ratio η and the amount ΔF ofchange in the frequency in a case where the film thickness H of theelectrode is set to 0.02λ.

Here, FIGS. 18A to 18F are diagrams illustrating the relationshipbetween the line occupancy ratio η and the second-order temperaturecoefficient β in a case where the film thickness H of the electrode isset to 0.025λ, and FIGS. 24A to 24F are diagrams illustrating therelationship between the line occupancy ratio η and the amount ΔF ofchange in the frequency in a case where the film thickness H of theelectrode is set to 0.025λ.

Here, FIGS. 19A to 19F are diagrams illustrating the relationshipbetween the line occupancy ratio η and the second-order temperaturecoefficient β in a case where the film thickness H of the electrode isset to 0.03λ, and FIGS. 25A to 25F are diagrams illustrating therelationship between the line occupancy ratio η and the amount ΔF ofchange in the frequency in a case where the film thickness H of theelectrode is set to 0.03λ.

Here, FIGS. 20A to 20F are diagrams illustrating the relationshipbetween the line occupancy ratio η and the second-order temperaturecoefficient β in a case where the film thickness H of the electrode isset to 0.035λ, and FIGS. 26A to 26F are diagrams illustrating therelationship between the line occupancy ratio η and the amount ΔF ofchange in the frequency in a case where the film thickness H of theelectrode is set to 0.035λ.

Although there are fine differences between the graphs of the diagrams(FIGS. 15A to 26F), it can be understood that the tendency of thechanges is similar to that of FIGS. 8A to 8I and 10A to 10I that aregraphs illustrating the relationship between the line occupancy ratio ηof only the quartz crystal substrate 30 and the second-order temperaturecoefficient β and the relationship between the line occupancy ratio ηand the amount ΔF of change in the frequency.

In other words, it can be stated that the advantage of this embodimentcan be accomplished for the propagation of the surface acoustic wave inthe single body of the quartz crystal substrate 30 excepting for theelectrode film.

Simulation was performed in a case where the range of the film thicknessH of the electrode was determined, and the depth G of the grooves waschanged for the range of η1 and η2 when the range of β was expanded upto |β|≦0.01 for each of two points η1 and η2 at which the second-ordertemperature coefficient β was zero. Here, as η1 and η2, a larger one ofη for which |β|≦0.01 is set as η1, and a smaller one of η for which|β|≦0.01 is set as η2. In addition, the quartz crystal substrate havingthe Euler angles of (0°, 123°, ψ) is used, and an angle is appropriatelyselected as ψ such that ΔF is the minimum.

FIG. 27A is a graph illustrating the relationship between η1 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.000λ<H≦0.005λ.Table 3 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 27A andthe values of β at the corresponding measurement points.

TABLE 3 Point G/λ η β a 0.0100 0.7100 −0.0098 b 0.0200 0.7100 −0.0099 c0.0300 0.7100 −0.0095 d 0.0400 0.7100 −0.0100 e 0.0500 0.7100 −0.0100 f0.0600 0.7100 −0.0098 g 0.0700 0.7100 −0.0099 h 0.0800 0.7100 0.0097 i0.0900 0.7100 −0.0100 j 0.0900 0.4200 0.0073 k 0.0800 0.5700 0.0086 l0.0700 0.5900 0.0093 m 0.0600 0.6150 0.0077 n 0.0500 0.6300 0.0054 o0.0400 0.6350 0.0097 p 0.0300 0.6500 0.0097 q 0.0200 0.6700 0.0074 r0.0100 0.7100 0.0091

From FIG. 27A and Table 3, it can be read that β satisfies theabove-described conditions within the area surrounded by the measurementpoints a to r for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.09λ in a case where the film thickness H of theelectrode is in the above-described range at η1.

FIG. 27B is a graph illustrating the relationship between η2 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.000λ<H≦0.005λ.Table 4 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 27B andthe values of β at the corresponding measurement points.

TABLE 4 Point G/λ η β a 0.0300 0.5900 0.0097 b 0.0400 0.5800 0.0097 c0.0500 0.5500 0.0054 d 0.0600 0.5200 0.0077 e 0.0700 0.4800 0.0093 f0.0800 0.4500 0.0086 g 0.0900 0.4000 0.0073 h 0.0900 0.1800 0.0056 i0.0800 0.3400 0.0093 j 0.0700 0.4100 0.0078 k 0.0600 0.4600 0.0094 l0.0500 0.4900 0.0085 m 0.0400 0.5200 0.0099 n 0.0300 0.5500 0.0098

From FIG. 27B and Table 4, it can be read that β satisfies theabove-described conditions within the area surrounded by measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.03λ≦G≦0.09λ in a case where the film thickness H of theelectrode is in the above-described range at η2.

FIG. 28A is a graph illustrating the relationship between η1 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.005λ<H≦0.010λ.Table 5 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 28A andthe values of β at the corresponding measurement points.

TABLE 5 Point G/λ η β a 0.0100 0.7700 −0.0099 b 0.0200 0.7400 −0.0100 c0.0300 0.7150 −0.0100 d 0.0400 0.7300 −0.0098 e 0.0500 0.7400 −0.0100 f0.0600 0.7300 −0.0098 g 0.0700 0.7300 −0.0100 h 0.0800 0.7300 −0.0100 i0.0800 0.5000 0.0086 j 0.0700 0.5700 0.0100 k 0.0600 0.6100 0.0095 l0.0500 0.6300 0.0100 m 0.0400 0.6350 0.0097 n 0.0300 0.6550 0.0070 o0.0200 0.6800 0.0100 p 0.0100 0.7600 0.0016

From FIG. 28A and Table 5, it can be read that β satisfies theabove-described conditions within the area surrounded by the measurementpoints a to p for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.08λ in a case where the film thickness H of theelectrode is in the above-described range at η1.

FIG. 28B is a graph illustrating the relationship between η2 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.005λ<H≦0.010λ.Table 6 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 28B andthe values of β at the corresponding measurement points.

TABLE 6 Point G/λ η β a 0.0200 0.6500 0.0090 b 0.0300 0.6100 0.0098 c0.0400 0.5700 0.0097 d 0.0500 0.5500 0.0040 e 0.0600 0.5200 0.0066 f0.0700 0.4700 0.0070 g 0.0700 0.3700 −0.0094 h 0.0600 0.4400 −0.0096 i0.0500 0.4800 −0.0096 j 0.0400 0.5200 −0.0095 k 0.0300 0.5500 −0.0099 l0.0200 0.5900 −0.0100

From FIG. 28B and Table 6, it can be read that β satisfies theabove-described conditions within the area surrounded by measurementpoints a to l for a case where the depth G of the grooves is in therange of 0.02λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η2.

FIG. 29A is a graph illustrating the relationship between η1 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.010λ<H≦0.015λ.Table 7 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 29A andthe values of β at the corresponding measurement points.

TABLE 7 Point G/λ η β A 0.0100 0.770 −0.0099 B 0.0200 0.760 −0.0099 C0.0300 0.760 −0.0099 D 0.0400 0.750 −0.0099 E 0.0500 0.750 −0.0099 F0.0600 0.750 −0.0099 G 0.0700 0.740 −0.0099 H 0.0800 0.740 −0.0098 I0.0800 0.340 0.0088 J 0.0700 0.545 0.0088 K 0.0600 0.590 0.0099 L 0.05000.620 0.0090 M 0.0400 0.645 0.0060 N 0.0300 0.670 0.0030 O 0.0200 0.7050.0076 P 0.0100 0.760 0.0010

From FIG. 29A and Table 7, it can be read that β satisfies theabove-described conditions within the area surrounded by the measurementpoints a to p for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.08λ in a case where the film thickness H of theelectrode is in the above-described range at η1.

FIG. 29B is a graph illustrating the relationship between η2 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.010λ<H≦0.015λ.Table 8 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 29B andthe values of β at the corresponding measurement points.

TABLE 8 Point G/λ η β a 0.0100 0.740 0.0099 b 0.0200 0.650 0.0090 c0.0300 0.610 0.0090 d 0.0400 0.570 0.0080 e 0.0500 0.540 0.0060 f 0.06000.480 0.0060 g 0.0700 0.430 0.0099 h 0.0700 0.3500 −0.0099 i 0.06000.4200 −0.0090 j 0.0500 0.4700 −0.0090 k 0.0400 0.5100 −0.0090 l 0.03000.5500 −0.0090 m 0.0200 0.6100 −0.0099 n 0.0100 0.7000 −0.0099

From FIG. 29B and Table 8, it can be read that β satisfies theabove-described conditions within the area surrounded by measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η2.

FIG. 30A is a graph illustrating the relationship between η1 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.015λ<H≦0.020λ.Table 9 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 30A andthe values of β at the corresponding measurement points.

TABLE 9 Point G/λ η β a 0.010 0.770 −0.0100 b 0.020 0.770 −0.0100 c0.030 0.760 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0100 f 0.0600.750 −0.0100 g 0.070 0.750 −0.0100 h 0.070 0.510 0.0100 i 0.060 0.5700.0099 j 0.050 0.620 0.0097 k 0.040 0.640 0.0096 l 0.030 0.660 0.0080 m0.020 0.675 0.0076 n 0.010 0.700 0.0010

From FIG. 30A and Table 9, it can be read that β satisfies theabove-described conditions within the area surrounded by the measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η1.

FIG. 30B is a graph illustrating the relationship between η2 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.015λ<H≦0.020λ.Table 10 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 30B andthe values of β at the corresponding measurement points.

TABLE 10 Point G/λ η β a 0.010 0.690 0.0010 b 0.020 0.640 0.0090 c 0.0300.590 0.0090 d 0.040 0.550 0.0080 e 0.050 0.510 0.0080 f 0.060 0.4700.0090 g 0.070 0.415 0.0100 h 0.070 0.280 −0.0100 i 0.060 0.380 −0.0090j 0.050 0.470 −0.0090 k 0.040 0.510 −0.0090 l 0.030 0.550 −0.0090 m0.020 0.610 −0.0100 n 0.010 0.680 −0.0100

From FIG. 30B and Table 10, it can be read that β satisfies theabove-described conditions within the area surrounded by measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η2.

FIG. 31A is a graph illustrating the relationship between η1 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.020λ<H≦0.025λ.Table 11 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 31A andthe values of β at the corresponding measurement points.

TABLE 11 Point G/λ η β a 0.010 0.770 −0.0100 b 0.020 0.770 −0.0100 c0.030 0.760 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0096 f 0.0600.760 −0.0100 g 0.070 0.760 −0.0100 h 0.070 0.550 0.0100 i 0.060 0.5450.0090 j 0.050 0.590 0.0097 k 0.040 0.620 0.0100 l 0.030 0.645 0.0100 m0.020 0.680 0.0070 n 0.010 0.700 0.0030

From FIG. 31A and Table 11, it can be read that β satisfies theabove-described conditions within the area surrounded by the measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η1.

FIG. 31B is a graph illustrating the relationship between η2 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.020λ<H≦0.025λ.Table 12 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 31B andthe values of β at the corresponding measurement points.

TABLE 12 Point G/λ η β a 0.010 0.690 0.0030 b 0.020 0.640 0.0090 c 0.0300.590 0.0090 d 0.040 0.550 0.0090 e 0.050 0.510 0.0080 f 0.060 0.4200.0090 g 0.070 0.415 0.0080 h 0.070 0.340 −0.0098 i 0.060 0.340 −0.0100j 0.050 0.420 −0.0100 k 0.040 0.470 −0.0100 l 0.030 0.520 −0.0093 m0.020 0.580 −0.0100 n 0.010 0.650 −0.0090

From FIG. 31B and Table 12, it can be read that β satisfies theabove-described conditions within the area surrounded by measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η2.

FIG. 32A is a graph illustrating the relationship between η1 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.025λ<H≦0.030λ.Table 13 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 32A andthe values of β at the corresponding measurement points.

TABLE 13 Point G/λ η β a 0.010 0.770 −0.0098 b 0.020 0.770 −0.0100 c0.030 0.770 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0099 f 0.0600.760 −0.0100 g 0.070 0.760 −0.0100 h 0.070 0.550 0.0080 i 0.060 0.5050.0087 j 0.050 0.590 0.0090 k 0.040 0.620 0.0100 l 0.030 0.645 0.0100 m0.020 0.680 0.0083 n 0.010 0.700 0.0052

From FIG. 32A and Table 13, it can be read that β satisfies theabove-described conditions within the area surrounded by the measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07, in a case where the film thickness H of theelectrode is in the above-described range at η1.

FIG. 32B is a graph illustrating the relationship between η2 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.025λ<H≦0.030λ.Table 14 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 32B andthe values of β at the corresponding measurement points.

TABLE 14 Point G/λ η β a 0.010 0.670 0.0052 b 0.020 0.605 0.0081 c 0.0300.560 0.0092 d 0.040 0.520 0.0099 e 0.050 0.470 0.0086 f 0.060 0.3950.0070 g 0.070 0.500 0.0080 h 0.070 0.490 −0.0100 i 0.060 0.270 −0.0100j 0.050 0.410 −0.0100 k 0.040 0.470 −0.0100 l 0.030 0.520 −0.0093 m0.020 0.580 −0.0099 n 0.010 0.620 −0.0090

From FIG. 32B and Table 14, it can be read that β satisfies theabove-described conditions within the area surrounded by measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η2.

FIG. 33A is a graph illustrating the relationship between η1 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.030λ<H≦0.035λ.Table 15 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 33A andthe values of β at the corresponding measurement points.

TABLE 15 Point G/λ η β a 0.010 0.770 −0.0100 b 0.020 0.770 −0.0098 c0.030 0.770 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0100 f 0.0600.760 −0.0100 g 0.070 0.760 −0.0100 h 0.070 0.550 0.0090 i 0.060 0.5000.0087 j 0.050 0.545 0.0090 k 0.040 0.590 0.0091 l 0.030 0.625 0.0080 m0.020 0.650 0.0083 n 0.010 0.680 0.0093

From FIG. 33A and Table 15, it can be read that β satisfies theabove-described conditions within the area surrounded by the measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η1.

FIG. 33B is a graph illustrating the relationship between η2 satisfyingthe above-described range of β and the depth G of the grooves in a casewhere the film thickness H of the electrode is set as 0.030λ<H≦0.035λ.Table 16 is a table that represents the coordinates (G/λ, η) of mainmeasurement points used for determining the range shown in FIG. 33B andthe values of β at the corresponding measurement points.

TABLE 16 Point G/λ η β a 0.010 0.655 0.0080 b 0.020 0.590 0.0081 c 0.0300.540 0.0092 d 0.040 0.495 0.0099 e 0.050 0.435 0.0090 f 0.060 0.3950.0061 g 0.070 0.500 0.0090 h 0.070 0.550 −0.0100 i 0.060 0.380 −0.0090j 0.050 0.330 −0.0100 k 0.040 0.410 −0.0095 l 0.030 0.470 −0.0099 m0.020 0.520 −0.0100 n 0.010 0.590 −0.0100

From FIG. 33B and Table 16, it can be read that β satisfies theabove-described conditions within the area surrounded by measurementpoints a to n for a case where the depth G of the grooves is in therange of 0.01λ≦G≦0.07λ in a case where the film thickness H of theelectrode is in the above-described range at η2.

In FIG. 35, the relationship between ψ acquired from η1 on the graphshown in FIGS. 34A to 34F and the depth G of the grooves is organized.The reason for selecting η1 is as described above. As shown in FIG. 35,even in a case where the film thickness of the electrode is changed,there is almost no difference in the angle of ψ, and it is understoodthat the optimal angle of ψ is changed in accordance with the change inthe depth G of the grooves. This also can be evidence representing thatthe ratio of change in the second-order temperature coefficient β thatis caused by the form of the quartz crystal substrate 30 is high.

As described above, the relationships between ψ at which thesecond-order temperature coefficient β=−0.01 (ppm/° C.²) and ψ at whichβ=+0.01 (ppm/° C.²) and the depth G of the grooves are acquired and areorganized in FIGS. 36 and 37. When the angle of ψ at which −0.01≦β≦+0.01is acquired from the graphs (FIGS. 35 to 37), a preferred range of theangle of ψ under the above-described conditions can be determined as43°<ψ<45°, and a more preferred range of the angle of ψ can bedetermined as 43.2°≦ψ≦44.2°.

In addition, simulation was performed for the range of ψ satisfying thecondition of |β|≦0.01 when the depth G of the grooves is changed in acase where the film thickness H of the electrode is changed. The resultsof the simulation are shown in FIGS. 38A to 44B. In addition, the quartzcrystal substrate having Euler angles of (0°, 123°, ψ) is used, and anangle is appropriately selected as ψ such that ΔF is the minimum.

FIG. 38A is a graph illustrating the range of ψ satisfying the conditionof |β|≦0.01 in a case where the film thickness H of the electrode is setas 0<H≦0.005λ. Here, a range interposed between a line joining plotsrepresenting maximum values of ψ and a broken line joining plotsrepresenting minimum values of ψ is the range that satisfies theabove-described condition.

When the range of a solid line and a broken line shown in FIG. 38A isapproximated as a polygon for the depth G of the grooves set in therange of 0.01≦G≦0.0695λ, it can be represented as FIG. 38B, andaccordingly, it can be stated that β satisfies the above-describedcondition in a range corresponding to the inner side of the polygondenoted by solid lines in FIG. 38B. When the range of the polygonrepresented in FIG. 38B is represented by an approximate equation, itcan be represented as Equations (14) and (15).ψ≦3.0×G/λ+43.92, wherein 0.0100λ≦G≦0.0695λ  (14)ψ≧−48.0×G/λ+44.35, wherein 0.0100λ≦G≦0.0695λ  (15)

FIG. 39A is a graph illustrating the range of ψ satisfying the conditionof |β|≦0.01 in a case where the film thickness H of the electrode is setas 0.005λ<H≦0.010λ. Here, a range interposed between a line joiningplots representing maximum values of ψ and a broken line joining plotsrepresenting minimum values of ψ is the range that satisfies theabove-described condition.

When the range of a solid line and a broken line shown in FIG. 39A isapproximated as a polygon for the depth G of the grooves set in therange of 0.01λ≦G≦0.0695λ, it can be represented as FIG. 39B, andaccordingly, it can be stated that β satisfies the above-describedcondition in a range corresponding to the inner side of the polygondenoted by solid lines in FIG. 39B. When the range of the polygonrepresented in FIG. 39B is represented by approximate equations, it canbe represented as Equations (16) and (17).ψ≦8.0×G/λ+43.60, wherein 0.0100λ≦G≦0.0695λ  (16)ψ≧−48.0×G/λ+44.00, wherein 0.0100λ≦G≦0.0695λ  (17)

FIG. 40A is a graph illustrating the range of ψ satisfying the conditionof |β|≦0.01 in a case where the film thickness H of the electrode is setas 0.010λ<H≦0.015λ. Here, a range interposed between a line joiningplots representing maximum values of ψ and a broken line joining plotsrepresenting minimum values of ψ is the range that satisfies theabove-described condition.

When the range of a solid line and a broken line shown in FIG. 40A isapproximated as a polygon for the depth G of the grooves set in therange of 0.01λ≦G≦0.0695λ, it can be represented as FIG. 40B, andaccordingly, it can be stated that β satisfies the above-describedcondition in a range corresponding to the inner side of the polygondenoted by solid lines in FIG. 40B. When the range of the polygonrepresented in FIG. 40B is represented by approximate equations, it canbe represented as Equations (18) and (19).ψ≦10.0×G/λ+43.40, wherein 0.0100λ≦G≦0.0695λ  (18)ψ≧−44.0×G/λ+43.80, wherein 0.0100λ≦G≦0.0695λ  (19)

FIG. 41A is a graph illustrating the range of ψ satisfying the conditionof |β|≦0.01 in a case where the film thickness H of the electrode is setas 0.015λ<H≦0.020λ. Here, a range interposed between a line joiningplots representing maximum values of ψ and a broken line joining plotsrepresenting minimum values of ψ is the range that satisfies theabove-described condition.

When the range of a solid line and a broken line shown in FIG. 41A isapproximated as a polygon for the depth G of the grooves set in therange of 0.01λ≦G≦0.0695λ, it can be represented as FIG. 41B, andaccordingly, it can be stated that β satisfies the above-describedcondition in a range corresponding to the inner side of the polygondenoted by solid lines in FIG. 41B. When the range of the polygonrepresented in FIG. 41B is represented by approximate equations, it canbe represented as Equations (20) and (21).ψ≦12.0×G/λ+43.31, wherein 0.0100λ≦G≦0.0695λ  (20)ψ≧−30.0×G/λ+44.40, wherein 0.0100λ≦G≦0.0695λ  (21)

FIGS. 42A and 42B are graphs illustrating the ranges of ψ satisfying thecondition of |β|≦0.01 in a case where the film thickness H of theelectrode is set as 0.020λ<H≦0.025λ. Here, a range interposed between aline joining plots representing maximum values of ψ and a broken linejoining plots representing minimum values of ψ is the range thatsatisfies the above-described condition.

When the range of a solid line and broken lines shown in FIG. 42A isapproximated as a polygon for the depth G of the grooves set in therange of 0.01λ≦G≦0.0695λ, it can be represented as FIG. 42B, andaccordingly, it can be stated that β satisfies the above-describedcondition in a range corresponding to the inner side of the polygondenoted by solid lines in FIG. 42B. When the range of the polygonrepresented in FIG. 42B is represented by approximate equations, it canbe represented as Equations (22) to (24).ψ≦14.0×G/λ+43.16, wherein 0.0100λ≦G≦0.0695λ  (22)ψ≧−45.0×G/λ+43.35, wherein 0.0100λ≦G≦0.0600λ  (23)ψ≧367.368×G/λ+18.608, wherein 0.0600λ≦G≦0.0695λ  (24)

FIG. 43A is a graph illustrating the range of ψ satisfying the conditionof |β|≦0.01 in a case where the film thickness H of the electrode is setas 0.025λ<H≦0.030λ. Here, a range interposed between a line joiningplots representing maximum values of ψ and a broken line joining plotsrepresenting minimum values of ψ is the range that satisfies theabove-described condition.

When the range of a solid line and a broken line shown in FIG. 43A isapproximated as a polygon for the depth G of the grooves set in therange of 0.01λ≦G≦0.0695λ, it can be represented as FIG. 43B, andaccordingly, it can be stated that β satisfies the above-describedcondition in a range corresponding to the inner side of the polygondenoted by solid lines in FIG. 43B. When the range of the polygonrepresented in FIG. 43B is represented by approximate equations, it canbe represented as Equations (25) to (27).ψ≦12.0×G/λ+43.25, wherein 0.0100λ≦G≦0.0695λ  (25)ψ≧−50.0×G/λ+43.32, wherein 0.0100λ≦G≦0.0500λ  (26)ψ≧167.692×G/λ+32.435, wherein 0.0500λ≦G≦0.0695λ  (27)

FIG. 44A is a graph illustrating the range of ψ satisfying the conditionof |β|≦0.01 in a case where the film thickness H of the electrode is setas 0.030λ<H≦0.035λ. Here, a range interposed between a line joiningplots representing maximum values of ψ and broken lines joining plotsrepresenting minimum values of ψ is the range that satisfies theabove-described condition.

When the range of a solid line and broken lines shown in FIG. 44A isapproximated as a polygon for the depth G of the grooves set in therange of 0.01λ≦G≦0.0695λ, it can be represented as FIG. 44B, andaccordingly, it can be stated that β satisfies the above-describedcondition in a range corresponding to the inner side of the polygondenoted by solid lines in FIG. 44B. When the range of the polygonrepresented in FIG. 44B is represented by approximate equations, it canbe represented as Equations (28) to (30).ψ≦12.0×G/λ+43.35, wherein 0.0100λ≦G≦0.0695λ  (28)ψ≧45.0×G/λ+42.80, wherein 0.0100λ≦G≦0.0500λ  (29)ψ≧186.667×G/λ+31.217, wherein 0.0500λ≦G≦0.0695λ  (30)

Next, FIG. 45 shows a change in the second-order temperature coefficientβ when swing is made by the angle of θ, that is, the relationshipbetween θ and the second-order temperature coefficient β. Here, a SAWdevice used for the simulation was a quartz crystal substrate in whichthe cut angle and the propagation direction of the SAW were representedas Euler angles (0, θ, ψ), the depth G of the grooves was 0.04λ, and thefilm thickness H of the electrode was 0.02λ. In addition, a value atwhich the absolute value of the second-order temperature coefficient βwas minimum within the above-described angle range was selected as ψbased on the set angle of θ. Furthermore, η was set to 0.6383 based onEquation (8) described above.

Under such conditions, from FIG. 45 that represents the relationshipbetween θ and the second-order temperature coefficient β, it can be readthat the absolute value of the second-order temperature coefficient β isin the range of 0.01 (ppm/° C.²) in a case where θ is within the rangeequal to or higher than 117° and equal to or lower than 142°.Accordingly, it can be stated that by following to set θ in the range of117°≦θ≦142° for the above-described set values, a SAW resonator 10having good frequency-temperature characteristics can be configured.

Tables 17 to 19 are shown as simulation data for proving therelationship between θ and the second-order temperature coefficient β.

TABLE 17 H/λ G/λ θ β % % ° ppm/° C.² 0.01 4.00 117 −0.009 0.01 4.00 1420.005 3.50 4.00 117 −0.009 3.50 4.00 142 −0.008

Table 17 is a table representing the relationship between θ and thesecond-order temperature coefficient β in a case where the filmthickness H of the electrode is changed and represents the values of thesecond-order temperature coefficient β at threshold values (117° and142°) of θ in a case where the film thickness H of the electrode is setto 0.01% λ and a case where the film thickness H of the electrode is setto 3.50% λ. In addition, the depth G of the grooves in this simulationis 4% λ. From Table 17, it can be read that, even in a case where thefilm thickness H of the electrode is changed (0≈0.01% λ or 3.5% λ thatis defined as a threshold value of the film thickness of the electrode)in the range of 117°≦θ≦142°, |β|≦0.01 is satisfied regardless of thethickness.

TABLE 18 H/λ G/λ θ β % % ° ppm/° C.² 2.00 1.00 117 −0.009 2.00 1.00 142−0.008 2.00 6.95 117 −0.009 2.00 6.95 142 −0.009

Table 18 is a table representing the relationship between θ and thesecond-order temperature coefficient β in a case where the depth G ofthe grooves is changed and represents the values of the second-ordertemperature coefficient β at threshold values (117° and 142°) of θ in acase where the depth G of the grooves is set to 1.00% λ and 6.95% λ. Inaddition, the film thickness H of the electrode in this simulation is2.00% λ. From Table 18, it can be read that, even in a case where thedepth G of the grooves is changed (1.00% λ or 6.95% λ that is defined asa threshold value of the depth G of the grooves) in the range of117°≦θ≦142°, |β|≦0.01 is satisfied regardless of the depth.

TABLE 19 H/λ G/λ θ β % % η ° ppm/° C.² 2.00 4.00 0.62 117 −0.010 2.004.00 0.62 142 −0.003 2.00 4.00 0.76 117 −0.009 2.00 4.00 0.76 142 −0.009

Table 19 is a table representing the relationship between θ and thesecond-order temperature coefficient β in a case where the lineoccupancy ratio η is changed and represents the values of thesecond-order temperature coefficient β at threshold values (117° and142°) of θ in a case where the line occupancy ratio η is set to 0.62 and0.76. In addition, the film thickness H of the electrode in thissimulation is 2.00% λ, and the depth G of the grooves is 4.00% λ. FromTable 19, it can be read that, even in a case where the line occupancyratio η is changed (η=0.62 and 0.76 are a minimum value and a maximumvalue of η in a case where the depth of the grooves is set to 4% λ inFIG. 31A representing the relationship between the line occupancy ratioη (η1) and the depth G of the grooves in the range of 0.020λ to 0.025λof the film thickness H of the electrode) in the range of 117°≦θ≦142°,|β|≦0.01 is satisfied regardless of the value.

FIG. 46 is a graph that represents the relationship between the angle ofφ and the second-order temperature coefficient β in a case where aquartz crystal substrate 30 represented by Euler angles (φ, 123°,43.77°) is used, the depth G of the grooves is set to 0.04λ, the filmthickness H of the electrode is set to 0.02λ, and the line occupancyratio η is set to 0.65.

From FIG. 46, although the second-order temperature coefficient β isless than −0.01 in a case where φ is −2° or +2°, it can be read that theabsolute value of the second-order temperature coefficient β is reliablywithin the range of 0.01 in a case where φ is in the range of −1.5° to+1.5°. Accordingly, by following to set   in the range of −1.5°≦φ≦+1.5°for the above-described set values or by more preferably setting φ inthe range of −1°≦φ≦+1°, a SAW resonator 10 having goodfrequency-temperature characteristics can be configured.

In the description presented above, the optimal ranges of φ, θ, and ψare derived in relation with the depth G of the grooves under a fixedcondition. In contrast to this, FIG. 47 shows a desirable relationshipbetween θ and ψ at which the amount of change is minimum in thefrequency for the range of −40° C. to +85° C., and approximate equationsthereof are acquired. As shown in FIG. 47, the angle of ψ changes inaccordance with the increase in the angle of θ so as to rise for drawinga third-order curve. In addition, in the example shown in FIG. 47, ψ ina case where θ=117° is 42.79°, and ψ in a case where θ=142° is 49.57°.In a case where such plots are represented as an approximate curve, acurve denoted by a broken line in FIG. 47 is formed, and Equation (31)can be represented as an approximate equation.ψ=1.19024×10⁻³×θ³−4.48775×10⁻¹×θ²+5.64362×10¹×θ−2.32327×10³±1.0  (31)

Accordingly, ψ can be determined by determining θ, and the range of ψ ina case where the range of θ is 117°≦θ≦142° can be set as42.79°≦ψ≦49.57°. In addition, the depth G of the grooves and the filmthickness H of the electrode in this simulation are G=0.04λ and H=0.02λ.

For the same reasons as described above, by configuring a SAW resonator10 based on various set conditions of this embodiment, a SAW resonator10 that can realize good frequency-temperature characteristicssatisfying a target value can be achieved.

In addition, according to the SAW resonator 10 of this embodiment, asrepresented in Equation (7) and FIGS. 15A to 26F, the film thickness Hof the electrode film is set in the range of 0<H≦0.035λ, and thefrequency-temperature characteristics are improved thereon. Differentlyfrom a general technology of improving the frequency-temperaturecharacteristics by forming the film thickness H to be extremely thick,improvement of the frequency-temperature characteristics is realizedwhile the environment-resistant characteristics being maintained. FIG.54 represents the relationship between the film thickness of theelectrode (the film thickness of the Al electrode) and the variation inthe frequency in a heat-cycle test. The result of the heat-cycle testshown in FIG. 54 is acquired after repeating one cycle eight times inwhich the SAW resonator is exposed for 30 minutes under the ambience of−55° C., then the ambient temperature is raised up to +125° C., and theSAW resonator is exposed for 30 minutes. From FIG. 54, the filmthickness H of the electrode is 0.06λ, and, in the range of the filmthickness H of the electrode of the SAW resonator 10 according to thisembodiment, it can be read that the frequency variation (F variation) isequal to or less than ⅓ of a case where grooves between the electrodefingers are not arranged. Here, in any plot shown in FIG. 54, H+G=0.06λ.

In addition, a high-temperature test was performed in which the SAWresonator manufactured under the same conditions as those represented inFIG. 54 was left for 1000 hours at the ambience of 125° C., and it waschecked that the amount of variation of the SAW resonator according tothis embodiment (four conditions of H=0.03λ and G=0.03λ, H=0.02λ andG=0.04λ, H=0.015λ and G=0.045λ, and H=0.01λ and G=0.05λ) in thefrequency before and after the test was equal to or less than ⅓ of thatof a general SAW resonator (H=0.06λ and G=0).

Under the above-described conditions, a SAW resonator 10 manufacturedunder the conditions in which H+G=0.067λ (the aluminum film thickness is2000 Å, the depth of the groove is 4700 Å), the line occupancy ratio ofthe IDT ηi=0.6, the line occupancy ratio of the reflector ηr=0.8, Eulerangles are (0°, 123°, 43.5°), the number of IDTs is 120, theintersection width is 40λ (λ=10 μm), the number of the reflectors (nearone side) is 72 (36), and there is no inclination angle of the electrodefingers (the direction of arrangement of the electrode fingers and thedirection of the phase velocity of the SAW coincide with each other) hasthe frequency-temperature characteristics as shown in FIG. 48.

FIG. 48 is acquired by plotting the frequency-temperaturecharacteristics in a case where the number of test pieces n=4. From FIG.48, it can be read that the amount ΔF of variation in the frequency inthe operating temperature range of the test pieces is suppressed to beequal to or less than about 20 ppm.

In this embodiment, the influence of the depth G of the grooves, thefilm thickness H of the electrode, and the like on thefrequency-temperature characteristics has been described. A depth (leveldifference) acquired by combining the depth G of the grooves and thefilm thickness H of the electrode has influences also on the staticcharacteristics such as an equivalent circuit constant and a CI valueand the Q value. For example, FIG. 49 is a graph showing therelationship between the level difference and the CI value in a casewhere the level difference is changed from 0.062λ to 0.071λ. From FIG.49, it can be read that the CI value converges when the level differenceis 0.067λ and is not improved (is not lowered) in a case where the leveldifference is increased therefrom.

The frequency, the equivalent circuit constant, and the staticcharacteristics of the SAW resonator 10 that represents thefrequency-temperature characteristics as shown in FIG. 48 are organizedin FIG. 50. Here, F is a frequency, Q is a Q value, γ is a capacitanceratio, the CI is a CI (Crystal Impedance) value, and M is a performanceindex (Figure of Merit).

In addition, FIG. 52 shows a graph used for comparing the relationshipbetween a level difference and a Q value in a general SAW resonator andthe SAW resonator 10 according to this embodiment. In FIG. 52, a graphdenoted by a thick line represents the characteristics of the SAWresonator 10 according to this embodiment, in which the grooves arearranged between electrode fingers, and the resonance of the stop bandupper end mode is used. In addition, a graph denoted by a thin linerepresents the characteristics of a general SAW resonator, in whichgrooves are not arranged between the electrode fingers, and theresonance of the stop band upper end mode is used. As is apparent fromFIG. 52, by arranging the grooves between the electrode fingers andusing the resonance of the stop band upper end mode, in the range inwhich the level difference (G+H) is equal to or larger than 0.0407λ(4.07% λ), a Q value higher than that of a case where the resonance ofthe stop band lower end mode is used without arranging grooves betweenthe electrode fingers can be acquired.

Basic data of the SAW resonator relating to the simulation is as below.

Basic Data of SAW Resonator 10 According to this Embodiment

H: 0.02λ

G: changing

IDT Line Occupancy ratio ηi: 0.6

Reflector Line Occupancy ratio ηr: 0.8

Euler Angles (0°, 123°, 43.5°)

Number: 120

Intersection Width: 40λ (λ=10 μm)

Number of Reflectors (near one side): 60

Inclination Angle of Electrode Fingers: None

Basic Data of General SAW Resonator

H: changing

G: zero

IDT Line Occupancy ratio ηi: 0.4

Reflector Line Occupancy ratio ηr: 0.3

Euler Angles (0°, 123°, 43.5°)

Number: 120

Intersection Width: 40λ (λ=10 μm)

Number of Reflectors (near one side): 60

Inclination Angle of Electrode Fingers: None

When FIGS. 50 and 52 are referred to so as to compare thecharacteristics of the SAW resonators, it can be understood how much theQ value of the SAW resonator 10 according to this embodiment becomeshigher. It is thought that the increase in the Q value is based on theimprovement of the effect of energy confinement, and the reason is asfollows.

In order to efficiently perform energy confinement of the surfaceacoustic wave excited by the stop band upper end mode, as shown in FIG.53, the upper end frequency ft2 of the stop band of the IDT 12 is setbetween the lower end frequency fr1 of the stop band of the reflector 20and the upper end frequency fr2 of the stop band of the reflector 20. Inother words, it may be set so as to satisfy the following relationship.fr1<ft2<fr2  (32)

Accordingly, the reflection coefficient Γ of the reflector 20 increasesfor the upper end frequency ft2 of the stop band of the IDT 12, and theSAW of the stop band upper end mode that is excited from the IDT 12 isreflected from the reflector 20 to the IDT 12 side with a highreflection coefficient. Then, the confinement of the energy of the SAWof the stop band upper end mode becomes stronger, and a resonator havinglow loss can be realized.

In contrast to this, in a case where the relationship among the upperend frequency ft2 of the stop band of the IDT 12 and the lower endfrequency fr1 of the stop band of the reflector 20, and the upper endfrequency fr2 of the stop band of the reflector 20 is set as “ft2<fr1”or “fr2<ft2”, the reflection coefficient Γ of the reflector 20 for theupper end frequency ft2 of the stop band of the IDT 12 decreases, and itis difficult to realize a state of strong energy confinement.

Here, in order to realize the state represented in Equation (32), thestop band of the reflector 20 needs to be shifted to the higherfrequency zone side than the stop band of the IDT 12. In particular, itcan be implemented by setting the period of the arrangement of theconductive strips 22 of the reflector 20 to be shorter than that of theelectrode fingers 18 of the IDT 12.

In addition, the implementation of a high Q value can be achieved bysetting the depth of the groove between the conductive strips 22 of thereflector 20 to be larger than that of the grooves between the electrodefingers of the IDT 12 or setting the film thickness of the conductivestrip 22 of the reflector 20 to be larger than that of the electrodefingers 18 of the IDT 12.

FIGS. 59 and 60 are diagrams showing other configuration examples of aSAW device according to an embodiment of the invention and are diagramsrepresenting partially enlarged cross-sections.

In the SAW resonator 10 shown in FIG. 59, the film thickness of theconductive strip 22 of the reflector 20 and the film thickness of theelectrode finger 18 of the IDT 12 are approximately the same. On theother hand, the depth of the groove 322 arranged between the conductivestrips 22 of the reflector 20 is larger than the depth of the groove 321arranged between the electrode fingers 18 of the IDT 12. Accordingly,the reflection characteristics of the reflector 20 are improved, and theelectromechanical coupling coefficient of the IDT 12 is improved. As aresult, the improvement of the Q value of the SAW resonator 10 and thereduction of the CI value can be achieved together with a high degree.

Here, the film thickness of the conductive strip 22 of the reflector 20is denoted by H_(mR), the depth of the groove 322 arranged between theconductive strips 22 is denoted by H_(gR), the film thickness of theelectrode finger 18 of the IDT 12 is denoted by H_(mT), and the depth ofthe groove 321 arranged between the electrode fingers 18 is denoted byH_(gT). In addition, since the film thickness of the electrode finger18, in consideration of the function thereof, can be regarded as alength (distance) from the bottom of the adjacent groove 321 to thesurface of the electrode finger 18, a value that is acquired by dividinga sum of the film thickness H_(mT) of the electrode finger 18 and thedepth H_(gT) of the groove 321 by the wavelength λ of the surfaceacoustic wave is defined as an “effective film thickness H_(T)/λ” of theelectrode finger 18. Similarly, since the film thickness of theconductive strip 22, in consideration of the function thereof, can beregarded as a length (distance) from the bottom of the adjacent groove322 to the surface of the conductive strip 22, a value that is acquiredby dividing a sum of the film thickness H_(mR) of the conductive strip22 and the depth H_(gR) of the groove 322 by the wavelength λ of thesurface acoustic wave is defined as an “effective film thicknessH_(R)/λ” of the conductive strip 22.

At this time, the SAW resonator 10 shown in FIG. 59 is configured so asto satisfy the relationship of Equation (33).H _(T) /λ<H _(R)/λ  (33)

As the SAW resonator 10 satisfies the above-described relationship, theeffective film thickness H_(R) of the conductive strip 22 increases, andthe reflection characteristics of the reflector 20 is improved.Accordingly, the energy confinement effect of the SAW of the stop bandupper end mode becomes more remarkable, whereby the Q value is furtherimproved. In addition, since the effective film thickness H_(T) of theelectrode finger 18 of the IDT 12 relatively decreases, theelectromechanical coupling coefficient of the IDT 12 can be increased,whereby the CI value can be further decreased.

Furthermore, in the SAW resonator 10 shown in FIG. 59, as describedabove, the film thickness H_(mR) of the conductive strip 22 of thereflector 20 and the film thickness H_(mT) of the electrode finger 18 ofthe IDT 12 are approximately the same (H_(mT)/λ=H_(mR)/λ). Accordingly,in order to satisfy Equation (33) described above, the relationshiprepresented in Equation (34) needs to be formed between the depth H_(gR)of the groove 322 arranged between the conductive strips 22 and thedepth H_(gT) of the groove 321 arranged between the electrode fingers18.H _(gT) /λ<H _(gR)/λ  (34)

On the other hand, although the SAW resonator 10 shown in FIG. 60 alsosatisfies the relationship represented in Equation (33) described above,the configuration thereof is slightly different from that shown in FIG.59.

In the SAW resonator 10 shown in FIG. 60, the depth H_(gR) of the groove322 arranged between the conductive strips 22 and the depth H_(gT) ofthe groove 321 arranged between the electrode finger 18 is approximatelythe same (H_(gT)/λ=H_(gR)/λ). Accordingly, in order to satisfy Equation(33) described above, the relationship represented in Equation (35)needs to be formed between the film thickness H_(mR) of the conductivestrip 22 of the reflector 20 and the film thickness H_(mT) of theelectrode finger 18 of the IDT 12.H _(mT) /λ<H _(mR)/λ  (35)

Also for the SAW resonator 10 shown in any of FIGS. 59 and 60,implementation of a high Q value and implementation of low CI value canbe achieved together.

In addition, in the case of the SAW resonator 10 shown in FIG. 59, sincethe film thicknesses of the conductive strip 22 and the electrode finger18 are the same, the manufacturing can be easily performed by forming aconductive film having a single film thickness when the conductive strip22 and the electrode finger 18 are formed.

On the other hand, in the case of the SAW resonator 10 shown in FIG. 60,since the depths of the groove 322 and the groove 321 are the same, themanufacturing can be easily performed by performing an etching processwith one type of conditions, for example, when the grooves 322 and thegrooves 321 are formed by using an etching method or the like. Here, thenumber of the conductive strips 22 of the reflector 20 is notparticularly limited. For example, the number of the conductive strips22 is preferably in the range of about 10 to 500 and is more preferablyin the range of about 20 to 400.

In addition, it is preferable that the film thicknesses H_(mR) of aplurality of the conductive strips 22 arranged in the reflector 20 areabout equal. This similarly applies to the film thicknesses H_(mT) of aplurality of the electrode fingers 18, the depths H_(gR) of a pluralityof the grooves 322, and the depths H_(gT) of a plurality of the grooves321. In order to satisfy Equation (33) described above, a SAW resonator10 in which H_(mT)/λ>H_(mR)/λ, and the relationship of H_(gT)/λ<H_(gR)/λis strong enough to reverse the magnitude relationship ofH_(mT)/λ>H_(mR)/λ may be configured. Similarly, a SAW resonator 10 inwhich H_(gT)/λ>H_(gR)/λ, and the relationship of H_(mT)/λ<H_(mR)/λ isstrong enough to reverse the magnitude relationship of H_(gT)/λ>H_(gR)/λmay be configured.

Furthermore, according to the FIG. 50, it can be stated that high figureof merit M is acquired in addition to the implementation of a high Qvalue. FIG. 51 is a graph showing the relationship between the impedanceZ of the SAW resonator, from which data shown in FIG. 50 is acquired,and the frequency. From FIG. 51, it can be read that there is no uselessspurious near the resonant point.

In this embodiment, the IDT 12 configuring the SAW resonator 10 isillustrated such that all the electrode fingers thereof intersect in analternating manner. However, the SAW resonator 10 according to anembodiment of the invention can have considerable advantages through thequartz crystal substrate only. Accordingly, even in a case where theelectrode fingers 18 of the IDT 12 are thinned out, the same advantagescan be acquired.

In addition, the groove 32 may be partially arranged between theelectrode fingers 18 or between the conductive strips 22 of thereflector 20. Particularly, since the center portion of the IDT 12 inwhich the vibration displacement is high has dominant influence on thefrequency-temperature characteristics, a structure may be employed inwhich the groove 32 is arranged only at the above-described centerportion. Even by employing such a structure, a SAW resonator 10 havinggood frequency-temperature characteristics can be configured.

In the above-described embodiment, as the material of the electrodefilm, Al or an alloy using Al as its main constituent is described to beused. However, the electrode film may be configured by using anothermetal material, as long as the metal material for which the sameadvantages as those of the above-described embodiment are acquired.

In addition, although the above-described embodiment is a one-terminalpair SAW resonator in which only one IDT is arranged, an embodiment ofthe invention can be applied to a two-terminal pair SAW resonator inwhich a plurality of the IDTs is arranged and can be applied as well toa dual mode SAW filter of a vertically-coupled type or a side-coupledtype or a multiple mode SAW filter.

FIG. 61A represents the level difference (H_(mT) H_(gT)) of theelectrode finger 18 of the IDT 12 in the horizontal axis and the Q valuein the vertical axis and is a graph representing the trend of change inthe Q value when the effective film thickness H_(T) is changed. FIG. 61Brepresents the level difference (H_(mT)+H_(gT)) in the horizontal axisand the electromechanical coupling coefficient in the vertical axis andis a graph representing the trend of change in the electromechanicalcoupling coefficient when the effective film thickness H_(T) is changed.

As is apparent from FIGS. 61A and 61B, in the SAW resonator 10, althoughthe Q value increases in accordance with the increase in the effectivefilm thickness H_(T), relatively the electromechanical couplingcoefficient decreases, whereby the CI value increases.

Thus, by configuring the SAW resonator 10 so as to satisfy Equation (33)described above, the effective thin film H_(T) is appropriatelydecreased, and the implementation of a high Q value and theimplementation of a low CI value can be achieved together with a highdegree.

In the example of FIGS. 61A and 61B, for example, when the effectivefilm thickness H_(T) of the electrode finger 18 is suppressed to be5.5%, the Q value is high as about 15000, and the electromechanicalcoupling coefficient is increased by about 0.056%, and it can be readthat the implementation of a high Q value and the implementation of alow CI value can be achieved together.

Second Embodiment

Next, a surface acoustic wave resonator according to a second embodimentof the invention will be described.

FIG. 62 is a diagram showing a SAW device according to the secondembodiment of the invention and is a diagram representing a partiallyenlarged cross-section.

Hereinafter, the second embodiment will be described, and a differencebetween the first and second embodiments will be focused, and thedescription of similar configurations will be omitted.

The SAW resonator 10 shown in FIG. 62 is similar to the SAW resonator 10shown in FIG. 59 except that the conductive strip of the reflector(reflection unit) 20 is omitted. In other words, the reflector 20 of theSAW resonator 10 shown in FIG. 62 is configured by a plurality ofgrooves 322 that are formed by depressing the surface of a quartzcrystal substrate 30. According to this embodiment, since the formationof the conductive strip in the reflector 20 is omitted, themanufacturing of the reflector 20 can be easily performed. In addition,the characteristic variation of the reflector 20 that is accompaniedwith the formation of the conductive strip can be suppressed.

Here, the depth H_(gR) of the groove 322, the film thickness H_(mT) ofthe electrode finger 18 of the IDT 12, and the depth H_(gT) of thegroove 321 arranged between the electrode fingers 18 satisfy thefollowing relationship.

When a value that is acquired by dividing a sum of the film thicknessH_(mT) of the electrode finger 18 and the depth H_(gT) of the groove 321by the wavelength λ of the surface acoustic wave is defined as an“effective film thickness H_(T)/λ” of the electrode finger 18, the SAWresonator 10 shown in FIG. 62 is configured so as to satisfy therelationship represented in Equation (36).H _(T) /λ<H _(gR)/λ  (36)

As the SAW resonator 10 satisfies the above-described relationship,although the conductive strip is omitted, the reflection characteristicsof the reflector 20 are improved, and the energy confinement effect ofthe SAW of the stop band upper end mode becomes more remarkable, wherebythe Q value is further improved. In addition, since the effective filmthickness H_(T) of the electrode finger 18 of the IDT 12 relativelydecreases, the electromechanical coupling coefficient of the IDT 12 canbe increased, whereby the CI value can be further decreased.

According to the SAW resonator 10 of this embodiment, the sameoperations and advantages as those of the SAW resonator 10 of the firstembodiment are acquired.

FIG. 63 is a diagram showing another configuration example of the SAWresonator 10 shown in FIG. 62.

The SAW resonator 10 shown in FIG. 63 has one groove 322, and the depthof the groove 322 is deeper than that shown in FIG. 62. Although thereis only one groove 322 having such a sufficient depth, the samereflection characteristics as those of a plurality of the grooves 322are acquired. Accordingly, it is only needed to manufacture thereflector 20 and form the groove 322 having a large depth, andtherefore, the degree of easiness in manufacturing can be remarkablyraised.

At this time, the depth H_(gR) of the groove 322 is preferably equal toor larger than 3λ and is more preferably equal to or larger than 6λ.Accordingly, although there is only one groove 322, the reflectioncharacteristics that are necessary and sufficient are acquired.

Third Embodiment

Next, a surface acoustic wave resonator according to a third embodimentof the invention will be described.

FIG. 64 is a diagram showing a SAW device according to the thirdembodiment of the invention and is a diagram representing a partiallyenlarged cross-section.

Hereinafter, the third embodiment will be described, and a differencebetween the first and third embodiments will be focused, and thedescription of similar configurations will be omitted.

The SAW resonator 10 shown in FIG. 64 is similar to the SAW resonator 10shown in FIG. 59 except that the groove arranged between the conductivestrips 22 of the reflector (reflection unit) 20 is omitted. In otherwords, the reflector 20 of the SAW resonator 10 shown in FIG. 64 isconfigured by a plurality of conductive strips 22 formed on the quartzcrystal substrate 30. According to this embodiment, since the formationof the groove in the reflector 20 is omitted, the manufacturing of thereflector 20 can be easily performed. In addition, the characteristicvariation of the reflector 20 that is accompanied with the formation ofthe groove can be suppressed.

Here, the film thickness H_(mR) of the conductive strip 22, the filmthickness H_(mT) of the electrode finger 18 of the IDT 12, and the depthH_(gT) of the groove 321 arranged between the electrode fingers 18satisfy the following relationship.

When a value that is acquired by dividing a sum of the film thicknessH_(mT) of the electrode finger 18 and the depth H_(gT) of the groove 321by the wavelength λ of the surface acoustic wave is defined as an“effective film thickness H_(T)/λ” of the electrode finger 18, the SAWresonator 10 shown in FIG. 64 is configured so as to satisfy therelationship represented in Equation (37).H _(T) /λ<H _(mR)/λ  (37)

As the SAW resonator 10 satisfies the above-described relationship,although the groove is omitted, the reflection characteristics of thereflector 20 are improved, and the energy confinement effect of the SAWof the stop band upper end mode becomes more remarkable, whereby the Qvalue is further improved. In addition, since the effective filmthickness H_(T) of the electrode finger 18 of the IDT 12 relativelydecreases, the electromechanical coupling coefficient of the IDT 12 canbe increased, whereby the CI value can be further decreased.

According to the SAW resonator 10 of this embodiment, the sameoperations and advantages as those of the SAW resonator 10 of the firstembodiment are acquired.

Fourth Embodiment

Next, a surface acoustic wave resonator according to a fourth embodimentof the invention will be described.

FIG. 65 is a diagram showing a SAW device according to the fourthembodiment of the invention and is a diagram representing a partiallyenlarged cross-section.

Hereinafter, the fourth embodiment will be described, and a differencebetween the first and fourth embodiments will be focused, and thedescription of similar configurations will be omitted.

The SAW resonator 10 shown in FIG. 65 is similar to the SAW resonator 10shown in FIG. 59 except that the reflector (reflection unit) 20 isconfigured by an end surface of the quartz crystal substrate 30.According to this embodiment, since the reflector 20 can be formed byonly forming an end surface having a high degree of smoothness in thequartz crystal substrate 30, the manufacturing of the reflector 20 canbe more easily performed. In addition, the formation of the groove andthe conductive strip is unnecessary, and accordingly, the characteristicvariation of the reflector 20 that is accompanied with such formationcan be suppressed, and the SAW resonator 10 can be miniaturized.

As shown in FIG. 65, end surfaces 30 a and 30 b of the quartz crystalsubstrate 30 are positioned so as to interpose the IDT 12 therebetween,and both end surfaces 30 a and 30 b are parallel to each other. Inaddition, both the end surfaces 30 a and 30 b are configured so as to beparallel to the electrode fingers 18 of the IDT 12.

From the viewpoint of the reflection characteristics, it is preferablethat the smoothness of both the end surfaces 30 a and 30 b is high, andboth the end surfaces 30 a and 30 b are perpendicular to the surface ofthe quartz crystal substrate 30.

In addition, the separation distance of each of the end surfaces 30 aand 30 b from the IDT 12 is set in accordance with the wavelength λ ofthe surface acoustic wave. For example, the separation distance is setso as to be multiples of λ/2 from the center of the electrode finger 18.

Surface Acoustic Oscillator

Next, a SAW oscillator according to an embodiment of the invention willbe described with reference to FIGS. 55A and 55B. The SAW oscillatoraccording to an embodiment of the invention, as shown in FIGS. 55A and55B, is configured by the above-described SAW resonator 10, an IC(integrated circuit) 50 that controls the driving by applying a voltageto the IDT 12 of the SAW resonator 10, and a package that houses theabove-described components. FIG. 55A is a plan view excluding a lid, andFIG. 55B is a cross-sectional view taken along line A-A shown in FIG.55A.

In the SAW oscillator 100 according to this embodiment, the SAWresonator 10 and the IC 50 are housed in a same package 56, andelectrode patterns 54 a to 54 g formed on a base plate 56 a of thepackage 56, the inter digital transducers 14 a and 14 b of the SAWresonator 10, and pads 52 a to 52 f of the IC 50 are connected throughmetal wires 60. Then, the cavity of the package 56 in which the SAWresonator 10 and the IC 50 are housed is sealed by a lid 58 withairtightness. By configuring as described above, the IDT 12 (see FIG.1A), the IC 50, and external mounting electrodes, which are not shown inthe figure, formed on the bottom face of the package 56 can beelectrically connected.

In addition to the high frequency of a reference clock due to recenthigh-speed information telecommunication, in accompaniment with theminiaturization of a casing that starts with a blade server, theinfluence of the internal heat generation increases, and an increase inthe operating temperature range and high precision, which are requiredfor an electronic device built inside, are demanded in the market, andfurthermore, a stable operation for a long period under an environmentfrom low temperature to high temperature is needed for a wireless basestation that is installed outdoor and the like therein. Accordingly,since a SAW oscillator according to an embodiment of the invention hasexcellent frequency-temperature characteristics in which the amount ofvariation in the frequency is about 20 (ppm) or less in the operatingtemperature range (the temperature range for use: −40° C. to +85° C.),the SAW oscillator is preferred in such a market.

Electronic Apparatus

Since a SAW resonator according to an embodiment of the invention or aSAW oscillator including the SAW resonator greatly realizes theenhancement of the frequency-temperature characteristics, it cancontribute to the implementation of various sensors (electronicapparatuses) having high reliability, for example, by being applied to apressure sensor that is disclosed in JP-A-2007-333500, JP-A-2007-93213,and the like, an acceleration sensor that is disclosed inJP-A-2008-286520 and the like, a rotation speed sensor that is disclosedin JP-A-2008-286521 and the like, or the like.

In addition, since a SAW resonator according to an embodiment of theinvention or a SAW oscillator including the SAW resonator greatlyrealizes the enhancement of the frequency-temperature characteristics,it can greatly contribute to the implementation of a product havingexcellent frequency-temperature characteristics and having superiorjitter characteristics and phase-noise characteristics for electronicapparatuses such as a cellular phone, a hard disk, a personal computer,a tuner that receives BS and CS broadcasts, an apparatus that processesa high-frequency signal propagating through a coaxial cable or anoptical signal propagating through an optical cable, a server/networkdevice that needs a high-frequency high-precision clock (low jitter andlow-phase nose) in a wide temperature range, and a radio communicationdevice, thereby greatly contributing to the improvement of thereliability and the quality of a system.

As described above, since a SAW resonator according to an embodiment ofthe invention has an inflection point within the operating temperaturerange (temperature range for use: −40° C. to +85° C.) represented inFIG. 48, the frequency-temperature characteristics of about 20 ppm orless in which the amount of variation in the frequency is extremelysmall, which is a third-order curve or a curve close to a third-ordercurve, can be realized.

FIG. 56A is a graph showing the frequency-temperature characteristics ofa SAW resonator that is disclosed in JP-A-2006-203408. Although thefrequency-temperature characteristics represent a third-order curve, asis shown, the inflection point is present in an area exceeding theoperating temperature range (temperature range for use: −40° C. to +85°C.). Accordingly, a second-order curve having an apex point of theupward convex as shown in FIG. 56B is substantially formed. Accordingly,the amount of variation in the frequency is 100 (ppm) that is extremelylarge.

In contrast to this, according to a SAW resonator of an embodiment ofthe invention, the amount of variation in the frequency corresponds to athird-order curve or a curve close to a third-order curve in theoperating temperature range, and accordingly, the amount of variation inthe frequency is dramatically decreased. The changes in the amount ofvariation in the frequency within the operating range in a SAW resonatorin which the IDT and the reflector are coated with a coating film areshown in FIGS. 57 and 58.

The example shown in FIG. 57 is a diagram showing the amount ofvariation in the frequency within the operating temperature range in acase where the electrode is coated with alumina as a protection film.From FIG. 57, it can be read that the amount of variation in thefrequency within the operating temperature range can be controlled to beequal or less than 10 (ppm).

Basic Data of SAW Resonator According to Example Shown in FIG. 57

H (material: aluminum): 2000 (Å)

G: 4700 (Å) (H+G=0.067)

IDT Line Occupancy ratio ηi: 0.6

Reflector Line Occupancy ratio ηr: 0.8

Number of Rotated ST-Cut Substrates within Plane of Euler Angles (0°,123°, 43.5°): 120

Intersection Width: 40λ (λ=10 μm)

Number of Reflectors (near one side): 36

Inclination Angle of Electrode Fingers: None

Film Thickness of Protection Film (Alumina): 400 (Å)

Second-Order Temperature Coefficient β=+0.0007 (ppm/° C.²)

The example shown in FIG. 58 is a diagram showing the amount ofvariation in the frequency within the operating temperature range in acase where the electrode is coated with SiO₂ as a protection film. FromFIG. 58, it can be read that the amount of variation in the frequencywithin the operating temperature range can be controlled to be equal orless than 20 (ppm).

Basic Data of SAW Resonator According to Example Shown in FIG. 58

H (material: aluminum): 2000 (Å)

G: 4700 (Å) (H+G=0.067)

IDT Line Occupancy ratio ηi: 0.6

Reflector Line Occupancy ratio ηr: 0.8

Number of Rotated ST-Cut Substrates within Plane of Euler Angles (0°,123°, 43.5°): 120

Intersection Width: 40λ (λ=10 μm)

Number of Reflectors (near one side): 36

Inclination Angle of Electrode Fingers: None

Film Thickness of Protection Film (SiO₂): 400 (Å)

Second-Order Temperature Coefficient β=+0.0039 (ppm/° C.²)

As above, although a surface acoustic wave resonator, a surface acousticwave oscillator, and an electronic apparatus according to embodiments ofthe invention have been described, the invention is not limited thereto.

For example, a surface acoustic wave resonator according to anembodiment of the invention may be acquired by combining theabove-described embodiments. For example, in the reflector (reflectionunit), an area in which only a conductive strip is formed, an area inwhich only a groove is formed, an area in which a conductive strip and agroove are formed, and the like may be mixed. Furthermore, asufficiently deep groove or a reflection end surface may be combinedtherewith.

What is claimed is:
 1. A surface acoustic wave resonator comprising: anIDT that is disposed on a quartz crystal substrate of Euler angles(−1.5°≦φ≦1.5°, 117°≦θ≦142°, ψ) and excites a surface acoustic waveresonant in an upper part of a stop-band of the IDT; and inter-electrodefinger grooves that are acquired by depressing the substrate locatedbetween electrode fingers configuring the IDT; and wherein, in a casewhere a wavelength of the surface acoustic wave is λ, and a depth of theinter-electrode finger grooves is G, 0.01λ≦G is satisfied, wherein, in acase where a line occupancy ratio of the IDT is η, the depth G of theinter-electrode finger grooves and the line occupancy ratio η satisfyone of the following relationships:−2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775, wherein 0.0100λ≦G≦0.0500λ, and−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775, wherein 0.0500λ<G≦0.0695λ,wherein, in a case where a film thickness of the electrode fingers ofthe IDT is H and a relationship of 0<H≦0.005λ is satisfied, thefollowing relationships are satisfied:ψ≦3.0°×G/λ+43.92°, wherein 0.0100λ≦G≦0.0695λ, andψ≧−48.0°×G/λ+44.35°, wherein 0.0100λ≦G≦0.0695λ, wherein, in a case wherea relationship of 0.005λ<H≦0.010λ is satisfied, the followingrelationships are satisfied:ψ≦8.0°×G/λ+43.60°, wherein 0.0100λ≦G≦0.0695λ, andψ≧−48.0°×G/λ+44.00°, wherein 0.0100λ≦G≦0.0695λ, wherein, in a case wherea relationship of 0.010λ<H≦0.015λ is satisfied, the followingrelationships are satisfied:ψ≦10.0°×G/λ+43.40°, wherein 0.0100λ≦G≦0.0695λ, andψ≧−44.0°×G/λ+43.80°, wherein 0.0100λ≦G≦0.0695λ, wherein, in a case wherea relationship of 0.015λ<H≦0.020λ is satisfied, the followingrelationships are satisfied:ψ≦12.0°×G/λ+43.31°, wherein 0.0100λ≦G≦0.0695λ, andψ≧−30.0°×G/λ+44.40°, wherein 0.0100λ≦G≦0.0695λ, wherein, in a case wherea relationship of 0.020λ<H≦0.025λ is satisfied, the followingrelationships are satisfied:ψ≦14.0°×G/λ+43.16°, wherein 0.0100λ≦G≦0.0695λ,ψ≧−45.0°×G/λ+43.35°, wherein 0.0100λ≦G≦0.0600λ, andψ≧367.368°×G/λ+18.608°, wherein 0.0600λ≦G≦0.0695λ, wherein, in a casewhere a relationship of 0.025λ<H≦0.030λ is satisfied, the followingrelationships are satisfied:ψ≦12.0°×G/λ+43.25°, wherein 0.0100λ≦G≦0.0695λ,ψ≧−50.0°×G/λ+43.32°, wherein 0.0100λ≦G≦0.0500λ, andψ≧167.692°×G/λ+32.435°, wherein 0.0500λ≦G≦0.0695λ, wherein, in a casewhere a relationship of 0.030λ<H≦0.035λ is satisfied, the followingrelationships are satisfied:ψ≦12.0°×G/λ+43.35°, wherein 0.0100λ≦G≦0.0695λ,ψ≧45.0°×G/λ+42.80°, wherein 0.0100λ≦G≦0.0500λ, andψ≧186.667°×G/λ+31.217°, wherein 0.0500λ≦G≦0.0695λ.
 2. The surfaceacoustic wave resonator according to claim 1, wherein a sum of the depthG of the inter-electrode finger grooves and the film thickness H of theelectrode fingers satisfies a relationship of 0.0407λ≦G+H.
 3. Thesurface acoustic wave resonator according to claim 1, wherein the Eulerangles ψ and θ satisfy a relationship of1.191°×10⁻³×θ³−4.490°×10⁻¹×θ²+5.646°×10¹×θ−2.324°×10³−1.0°≦ψ≦1.191°×10⁻³×θ³−4.490°×10⁻¹×θ²+5.646°×10¹×θ−2.324°×10³+1.0°.4. A surface acoustic wave oscillator comprising: the surface acousticwave resonator according to claim 1; and an IC that is used for drivingthe IDT.
 5. An electronic apparatus comprising: the surface acousticwave resonator according to claim
 1. 6. The surface acoustic waveresonator according to claim 1, wherein the line occupancy ratio ηsatisfies a relationship of−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)−0.04≦η≦−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)+0.04.7. The surface acoustic wave resonator according to claim 6, wherein asum of the depth G of the inter-electrode finger grooves and the filmthickness H of the electrode fingers satisfies a relationship of0.0407λ≦G+H.
 8. The surface acoustic wave resonator according to claim6, wherein the Euler angles ψ and θ satisfy a relationship of1.191°×10⁻³×θ³−4.490°×10⁻¹×θ²+5.646°×10¹×θ−2.324°×10³−1.0°≦ψ≦1.191°×10⁻³×θ³−4.490°×10⁻¹×θ²+5.646°×10¹×θ−2.324°×10³+1.0°.9. A surface acoustic wave oscillator comprising: the surface acousticwave resonator according to claim 6; and an IC that is used for drivingthe IDT.
 10. An electronic apparatus comprising: the surface acousticwave resonator according to claim 6.